8.1 Simplify Expressions with Roots

Содержание
  1. Squaring Removes Any Negative
  2. Even Exponents of Negative Numbers
  3. Example: What is (−1)97 ?
  4. Example: What is (−2)6 ?
  5. Roots of Negative Numbers
  6. Example: What is the value of x here: x2 = −1
  7. How Many of The Roots are Positive?
  8. Complex Roots
  9. Improving the Number of Positive Roots
  10. How Many of The Roots are Negative?
  11. But remember to reduce it because there may be Complex Roots!
  12. Total Number of Roots
  13. What we Know
  14. Must Have a Constant Term
  15. Example: 2x4 + 3x2 − 4x
  16. nth Root
  17. The nth Root
  18. The nth Root Symbol
  19. Using it
  20. Why «Root» … ?
  21. Properties
  22. Multiplication and Division
  23. Example:
  24. Addition and Subtraction
  25. Exponents vs Roots
  26. Example:
  27. nth Root of a-to-the-nth-Power
  28. nth Root of a-to-the-mth-Power
  29. Example:
  30. Example:
  31. Learning Objectives
  32. Simplify Expressions with Higher Roots
  33. nth Root of a Number
  34. Properties of a n a n
  35. Simplifying Odd and Even Roots
  36. Use the Product Property to Simplify Expressions with Higher Roots
  37. Simplified nth Root
  38. Product Property of nth Roots
  39. Use the Quotient Property to Simplify Expressions with Higher Roots
  40. Quotient Property of nth Roots
  41. Add and Subtract Higher Roots
  42. Section 9.7 Exercises
  43. Practice Makes Perfect
  44. Everyday Math
  45. Writing Exercises
  46. Self Check
  47. Learning Objectives
  48. Expressing Square Roots of Negative Numbers as Multiples of i
  49. Imaginary and Complex Numbers
  50. Expressing an Imaginary Number in Standard Form
  51. Plotting a Complex Number on the Complex Plane
  52. Plotting a Complex Number on the Complex Plane
  53. Adding and Subtracting Complex Numbers
  54. Complex Numbers: Addition and Subtraction
  55. Adding Complex Numbers
  56. Multiplying Complex Numbers
  57. Multiplying a Complex Number by a Real Number
  58. Multiplying a Complex Number by a Real Number
  59. Multiplying Complex Numbers Together
  60. Multiplying a Complex Number by a Complex Number
  61. Dividing Complex Numbers
  62. The Complex Conjugate
  63. Finding Complex Conjugates
  64. Dividing Complex Numbers
  65. Substituting a Complex Number into a Polynomial Function
  66. Substituting an Imaginary Number in a Rational Function
  67. Simplifying Powers of i
  68. Simplifying Powers of i i
  69. 3.1 Section Exercises
  70. Verbal
  71. Algebraic
  72. Graphical
  73. Numeric
  74. Technology
  75. Extensions
  76. Learning Objectives
  77. Simplify Expressions with Roots
  78. Square and Square Root of a number
  79. Square Root Notation
  80. nth Root of a Number
  81. Properties of a n a n
  82. Estimate and Approximate Roots
  83. Simplify Variable Expressions with Roots
  84. Simplifying Odd and Even Roots
  85. Section 8.1 Exercises
  86. Practice Makes Perfect
  87. Writing Exercises
  88. Self Check

Squaring Removes Any Negative

«Squaring» means to multiply a number by itself.

  • Squaring a positive number gets a positive result: (+5) × (+5) = +25
  • Squaring a negative number also gets a positive result: (−5) × (−5) = +25

Because a negative times a negative gives a positive. So:

5x5 = -5x-5

Square root of square

Oh no! We started with minus 3 and ended with plus 3.

When we square a number, then take the square root, we may not end up with the number we started with!

In fact we end up with the absolute value of the number:

That also happens for all even (but not odd) Exponents.

Even Exponents of Negative Numbers

An even exponent always gives a positive (or 0) result.

That simple fact can make our life easier:

(−1)1 = −1

(−1)2 = (−1) × (−1) = +1

(−1)3 = (−1) × (−1) × (−1) = −1

(−1)4 = (−1) × (−1) × (−1) × (−1) = +1

Do you see the −1, +1, −1, +1 pattern?

So we can «shortcut» some calculations, like:

Example: What is (−1)97 ?

97 is odd, so:

(−1)97 = −1

Example: What is (−2)6 ?

26 = 64, and 6 is even, so:

(−2)6 = +64

Roots of Negative Numbers

Example: What is the value of x here: x2 = −1

1 × 1 = +1

(−1) × (−1) = +1

We can’t get −1 for an answer!

It seems impossible!

Well, it is impossible using Real Numbers.

But we can do it using Imaginary Numbers.

In other words:

This is true for all even roots:

An Even Root of a Negative Number is Not Real

So just be careful when taking square roots, 4th roots, 6th roots, etc.

1742, 3998, 459, 3999, 460, 1743, 1093, 4000, 1094, 4001

A special way of telling how many positive and negative roots a polynomial has.

A Polynomial looks like this:

Polynomials have «roots» (zeros), where they are equal to 0:

polynomial roots
Roots are at x=2 and x=4
It has 2 roots, and both are positive (+2 and +4)

Let me show you with an example:

Example: 4x + x2 − 3x5 − 2

How Many of The Roots are Positive?

First, rewrite the polynomial from highest to lowest exponent (ignore any «zero» terms, so it does not matter that and are missing):

−3x5 + x2 + 4x − 2

Then, count how many times there is a change of sign (from plus to minus, or minus to plus):

Rule of Signs

The number of sign changes is the maximum number of positive roots

There are 2 changes in sign, so there are at most 2 positive roots (maybe less).

So there could be 2, or 1, or 0 positive roots ?

Complex Roots

There might also be complex roots.

Complex Number

Complex Roots always come in pairs!

Complex Conjugate Pairs

Always in pairs? Yes. So we either get:

  • no complex roots,
  • 2 complex roots,
  • 4 complex roots,
  • etc

Improving the Number of Positive Roots

So in our example from before, instead of 2 positive roots there might be 0 positive roots:

Number of Positive Roots is 2, or 0

This is the general rule:

The number of positive roots equals the number of sign changes, or a value less than that by some multiple of 2

Example: If the maximum number of positive roots was 5, then there could be 5, or 3 or 1 positive roots.

How Many of The Roots are Negative?

Rule of Signs

And then we need to work out the signs:

  • 3(−x)5 becomes 3x5
  • (−x)2 becomes x2 (no change in sign)
  • 4(−x) becomes 4x

So we get:

+3x5 + x2 − 4x − 2

The trick is that only the odd exponents, like 1,3,5, etc will reverse their sign.

Now we just count the changes like before:

Rule of Signs

One change only, so there is 1 negative root.

But remember to reduce it because there may be Complex Roots!

Total Number of Roots

On the page Fundamental Theorem of Algebra we explain that a polynomial will have exactly as many roots as its degree (the degree is the highest exponent of the polynomial).

Rule of Signs

So we know one more thing: the degree is 5 so there are 5 roots in total.

What we Know

OK, we have gathered lots of info. We know all this:

  • positive roots: 2, or 0
  • negative roots: 1
  • total number of roots: 5

So, after a little thought, the overall result is:

  • 5 roots: 2 positive, 1 negative, 2 complex (one pair), or
  • 5 roots: 0 positive, 1 negative, 4 complex (two pairs)

And we managed to figure all that out just based on the signs and exponents!

Must Have a Constant Term

One last important point:

Before using the Rule of Signs the polynomial must have a constant term (like «+2» or «−5»)

If it doesn’t, then just factor out x until it does.

Example: 2x4 + 3x2 − 4x

No constant term! So factor out «x»:

x(2x3 + 3x − 4)

This means that x=0 is one of the roots.

Now do the «Rule of Signs» for:

2x3 + 3x − 4

Count the sign changes for positive roots:

Rule of Signs
There is just one sign change,
So there is 1 positive root

And the negative case (after flipping signs of odd-valued exponents):

Rule of Signs
There are no sign changes,
So there are no negative roots

The degree is 3, so we expect 3 roots. There is only one possible combination:

  • 3 roots: 1 positive, 0 negative and 2 complex

And now, back to the original question:

2x4 + 3x2 − 4x

  • 4 roots: 1 zero, 1 positive, 0 negative and 2 complex

489, 490, 1130, 1131, 2420, 2421, 4023, 4024, 4025, 4026

Historical Note: The Rule of Signs was first described by René Descartes in 1637, and is sometimes called Descartes’ Rule of Signs.

nth Root

The «nth Root» used n times in a multiplication gives the original value

Instead of talking about the «4th», «16th», etc, we can just say the «».

The nth Root

  • The «2nd» root is the square root
  • The «3rd» root is the cube root
  • etc!

So it is the general way of talking about roots
(so it could be 2nd, or 9th, or 324th, or whatever)

The nth Root Symbol

nth root symbol

This is the special symbol that means «nth root»,
it is the «radical» symbol (used for square roots) with a little n to mean nth root.

Using it

We could use the nth root in a question like this:

Question: What is «n» in this equation?

= 5

Answer: I just happen to know that 625 = 54 , so the 4th root of 625 must be 5:

= 5

Or we could use «n» because we want to say general things:

Example: When n is odd then  

= a

  (we talk about this later).

Why «Root» … ?

Properties

Now we know what an nth root is, let us look at some properties:

Multiplication and Division

We can «pull apart» multiplications under the root sign like this:


=

×


(Note: if n is even then a and b must both be ≥ 0)

This can help us simplify equations in algebra, and also make some calculations easier:

Example:


=
=

×
=
4

So the cube root of 128 simplifies to 4 times the cube root of 2.

It also works for division:


=


(a≥0 and b>0)
Note that b cannot be zero, as we can’t divide by zero


=

=
1/4

So the cube root of 1/64 simplifies to just one quarter.

Addition and Subtraction

But we cannot do that kind of thing for additions or subtractions!

no!  

+

no!  
a − b

no!  
an + bn
a + b

Example: Pythagoras’ Theorem says

So we calculate c like this:

c = a2 + b2

Which is not the same as c = a + b , right?

It is an easy trap to fall into, so beware.

It also means that, unfortunately, additions and subtractions can be hard to deal with when under a root sign.

Exponents vs Roots

An exponent on one side of «=» can be turned into a root on the other side of «=»:

If  an = b  then  a =

Note: when n is even then b must be ≥ 0

Example:

54 = 625  so  5 =

nth Root of a-to-the-nth-Power

Square root of square

Did you see that −3 became +3 ?

Here it is in a little table:

nth Root of a-to-the-mth-Power

What happens when the exponent and root are different values (m and n)?

Well, we are allowed to change the order like this:

=

So this:    nth root of (a to the power m)
becomes  (nth root of a) to the power m

Example:

=

= 32
= 9

Easier than squaring 27 then taking a cube root, right?

=

The new exponent is the fraction which may be easier to solve.

Example:

=
= 42
= 16

This works because the nth root is the same as an exponent of (1/n)

=

You might like to read about Fractional Exponents to find out why!

318, 2055, 319, 317, 1087, 2056, 1088, 2057, 3159, 3160

252. Evolution is the opposite of involution. One is finding a power of a quantity, by multiplying it into itself. The other is finding a root, by resolving a quantity into equal factors. A quantity is resolved into any number of equal factors, by dividing its index into as many equal parts; (Art. 241.)

Divide the index of the quantity by the number expressing the root to be found.

Or, place over the quantity the radical sign belonging to the required root.

1. Thus the cube root of a6 is a2. For a2.a2.a2 = a6.

Here 6, the index of the given quantity, is divided by 3, the number expressing the cube root.

2. The cube root of a or a1, is a1/3 or 3√.

For a1/3.a1/3.a1/3r or 3√.3√.3√ = a. (Arts 239,242).

3. The 5th root of ab, is (ab)1/5 or 5√.

4. The n-th root of a2 is a2/n or n
5. The 7th root of 2d — x, is (2d — x)1/7 or 72d — x.

6. The cube root of a1/2, is a1/6. (Art. 160.)

7. The 4th root of a-1 is a-1/4.

8. The cube root of a2/3 is a2/9.

9. The n-th root of xm, is xm/n.

253. According to the rule just given, the cube root of the square root is found, by dividing the index 1/2 by 3, as in example 6th. But instead of dividing by 3, we may multiply by 1/3. For (1/2):3 = (1/2):(3/1) = (1/2).(1/3). (Art. 159.)

So (1/m):n = (1/m).(1/n). Therefore the roth root of the n-th root of a is equal to a(1/n).(1/m).

That is, (a1/n)1/m = a(1/n).(1/m) = a1/mn.

Here the two fractional indices are reduced to one by multiplication.

It is sometimes necessary to reverse this process; to resolve an index into two factors.

It may be necessary to observe, that resolving the index into factors, is not the same as resolving the quantity into factors. The latter is effected, by dividing the index into parts.

254. The rule in Art. 252, may be applied to every case in evolution. But when the quantity whose root is to be found, is composed of several factors, there will frequently be an advantage in taking the root of each of the factors separately.

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This is done upon the principle that the root of the product of severed factors, is equal to the product of their roots.

Thus √ = √.√. For each member of the equation if involved, will give the same power.

The square of √ is ab. (Art. 237.)

The square of √.√, is √.√.√.√. (Art. 99).

But √.√ = a. (Art. 237.) And √.√ = b.

Therefore the square of √.√ = √.√.√.√ = ab, which is also the √.

On the same principle, (ab)1/n =a1/n.b1/n.

When, therefore, a quantity consists of several factors, we may either extract the root of the whole together; or we may find the root of the factors separately, and then multiply them into each other.

Ex. 1. The cube root of xy, is either (xy)1/3 or x1/3.y1/3.

2. The 5th root of 3y, is 5√ or 5√.5√.

3. The 6th root of abh, is (abh)1/6, or a1/6b1/6h1/6.

4. The cube root of 8b, is (8b)1/3, or 2b1/3.

5. The n-th root of xny, is (xny)1/n or xy1/n.

255. The boot of a fraction is equal to the root of the numerator divided by the root of the denominator.

1. Thus the square root of a/b = a1/2/b1/2. For (a1/2/b1/2).(a1/2/b1/2) = a/b.

2. The square root of x/ay, √/√.

3. √ = √/√

256. For determining what sign to prefix to a root, it is important to observe, that

An odd root of any quantity has the same sign as the quantity itself.

An even boot of an affirmative quantity is ambiguous.


An even root of a negative quantity is impossible.

That the 3d, 5th, 7th, or any other odd root of a quantity must have the same sign as the quantity itself, is evident from Art. 215.

257. But an even root of an affirmative quantity may be either affirmative or negative. For, the quantity may be produced from the one, as well as from the other. (Art 215.)

Thus the square root of a2 is +a or -a.

An even root of an affirmative quantity is, therefore, said to be ambiguous, and is marked with both + and -.

Thus the square root of 3b, is ±√.

The 4th root of x, is ±x1/4.

The ambiguity does not exist, however, when, from the nature of the case, or a previous multiplication, it is known whether the power has actually been produced from a positive or from a negative quantity.

258. But no even root of a negative quantity can be found. The square root of-a2 is neither -a nor +a.

For +a.+a = +a2.

And -a.-a = +a2 also.

An even root of a negative quantity is, therefore, said to be impossible or imaginary.

There are purposes to be answered, however, by applying the radical sign to negative quantities. The expression √ is often to be found in algebraic processes. For, although we are unable to assign it a rank, among either positive or negative quantities; yet we know that when multiplied into itself, its product is -a, because √ is by notation a root of -a, that is, a quantity which multiplied into itself produces -a.

This may, at first view, seem to be an exception to the general rule that the product of two negatives is affirmative. But it is to be considered, that √ is not itself a negative quantity, but the root of a negative quantity.

The mark of subtraction here, must not be confounded with that which is prefixed to the radical sign. The expression √ is not equivalent to -√. The former is a root of -a; but the latter is a root of+a:

For -√.-√ = √ = a.

The root of -a, however, may be ambiguous. It may be either +√, or -√.

One of the uses of imaginary expressions is to indicate an impossible or absurd supposition in the statement of a problem. Suppose it be required to divide the number 14 into two such parts, that their product shall be 60. If one of the parts be x, the other will be 14 — x. And by the supposition,

         x.(14 — x) = 60, or 14x — x2 = 60.

As the value of x is here found to contain an imaginary expression, we infer that there is an inconsistency in the statement of the problem: that the number 14 cannot be divided into any two parts whose product shall be 60.

259. The methods of extracting the roots of compound quantities are to be considered in a future section. But there is one class of these, the squares of binomial and residual quantities, which it will be proper to attend to in this place. It has been shown (Art. 210,) that the square of a binomial quantity consists of three terms, two of which are complete powers, and the other is a double product of the roots of these powers. The square of a + b, for instance, is

         a2 + 2ab + b2,

two terms of which, a2 and b2 are complete powers and 2ab is twice the product of a into b, that is, the root of a2 into the root of b2.

Whenever, therefore, we meet with a quantity of this description, we may know that its square root is a binomial; and this may be found, by taking the root of the two terms which are complete powers, and connecting them by the sign +. The other term disappears in the root. Thus, to find the square root of

         x2 + 2xy + y2,

take the root of x2, and the root of y2 and connect them by the sign +. The binomial root will then be x + y.

In a residual quantity, the double product has the sign — prefixed, instead of +. The square of a — b, for instance, is a2 — 2ab + b2 (Art. 210.) And to obtain the root of a quantity of this description, we have only to take the roots of the two complete powers, and connect them by the sign -. Thus the square root of x2 -2xy + y2 is x — y. Hence,

260. To extract a binomial or residual square root, take the roots of the two terms which are complete powers, and connect them by the sign which is prefixed to the other term.

Ex. 1. To find the root of x2 + 2x + 1.

The two terms which are complete powers are x2 and 1.

The roots are x and 1. (Art. 244.)

The binomial root is, therefore, x + 1.

2. The square root of a2 + a + 1/4, is a + 1/2. (Art. 220.)

3. The square root of a2 + ab + b2/4, is a + b/2.

4. The square root of a2 + 2ab/c + b2/c2, a + b/c.

261. A root whose value cannot be exactly expressed in numbers, is called a surd.

Thus √ is a surd, because the square root of 2 cannot be expressed in numbers, with perfect exactness.

In decimals, it is 1.41421356 nearly.

But though we are unable to assign the value of such a quantity when taken alone, yet by multiplying it into itself, or by combining it with other quantities, we may produce expressions whose value can be determined. There is, therefore, a system of rules generally appropriated to surds. But as all quantities whatever, when under the same radical sign, or having the same index, may be treated in nearly the same manner; it will be most convenient to consider them together, under the general name of Radical Quantities; understanding by this term, every quantity which is found under a radical sign, or which has a fractional index.

262. Every quantity which is not a surd, is said to be rational. But for the purpose of distinguishing between radicals and other quantities, the term rational will be applied, in this section, to those only which do not appear under a radical sign, and which have not a fractional index.

Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with higher roots
  • Use the Product Property to simplify expressions with higher roots
  • Use the Quotient Property to simplify expressions with higher roots
  • Add and subtract higher roots


Before you get started, take this readiness quiz.

Simplify: .

If you missed this problem, review Example 6.18.


Simplify: .

If you missed this problem, review Example 6.22.


Simplify: .

If you missed this problem, review Example 6.59.

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

We write:We say:n2nsquaredn3ncubedn4nto the fourthn5nto the fifthWe write:We say:n2nsquaredn3ncubedn4nto the fourthn5nto the fifth

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from . See Figure 9.4.

This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing,  “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.


First through fifth powers of integers from −5−5 to 5.5.

Notice the signs in Figure 9.4. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of below to help you see this.

This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.

Earlier in this chapter we defined the square root of a number.

Ifn2=m,thennis a square root ofm.Ifn2=m,thennis a square root ofm.

And we have used the notation to denote the principal square root. So always.

We will now extend the definition to higher roots.

nth Root of a Number

If , then is an nth root of a number .

The principal nth root of is written .

        n is called the of the radical.

We do not write the index for a square root. Just like we use the word ‘cubed’ for , we use the term ‘cube root’ for .

We refer to Figure 9.4 to help us find higher roots.

Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of

a
n

a
n


When is an even number and

  • a≥0a≥0, then anan is a real number
  • a<0a<0, then anan is not a real number

When is an odd number, is a real number for all values of .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .

When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.

The odd root of a number can be either positive or negative. We have seen that .

But the even root of a non-negative number is always non-negative, because we take the principal nth root.

Suppose we start with .

Simplifying Odd and Even Roots

For any integer ,

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of .

Simplified nth Root

is considered simplified if has no factors of .

We will generalize the Product Property of Square Roots to include any integer root .

Product Property of nth Roots

when and are real numbers and for any integer


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.


Simplify: .


Simplify: .


Simplify: .

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Quotient Property of nth Roots

when anandbnare real numbers,b≠0,and for any integern≥2anandbnare real numbers,b≠0,and for any integern≥2


Simplify: .


Simplify: .


Simplify: .

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.


Simplify: .


Simplify: .


Simplify: .

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of .

Radicals with the same index and same radicand are called .

Like radicals have the same index and the same radicand.

  • 942x4942x4 and −242×4−242×4 are like radicals.
  • 5125x35125x3 and 6125y36125y3 are not like radicals. The radicands are different.
  • 21000q521000q5 and −41000q4−41000q4 are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add and the result is .


Simplify: .


Simplify: .


Simplify: .

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .


Simplify: .

Section 9.7 Exercises

Practice Makes Perfect

Simplify Expressions with Higher Roots

Use the Product Property to Simplify Expressions with Higher Roots

Use the Quotient Property to Simplify Expressions with Higher Roots

Add and Subtract Higher Roots

125

d

15

3

125

d

15

3

128

x
8

5

2

x
2

5

128

x
8

5

2

x
2

5

96

r

11

s
3

5

96

r

11

s
3

5

128

u
7

v
3

6

128

u
7

v
3

6

81

3

192

3

81

3

192

3

512

4

32

4

512

4

32

4

64

a

10

3

−216

a

12

3

64

a

10

3

−216

a

12

3

486

u
7

4

+

768

u
3

4

486

u
7

4

+

768

u
3

4

Everyday Math

Population growth The expression models the growth of a mold population after generations. There were 10 spores at the start, and each had offspring. So is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression to determine the number of offspring of each spore.

Spread of a virus The expression models the spread of a virus after cycles. There were three people originally infected with the virus, and each of them infected people. So is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression to determine the number of people each person infected.

Writing Exercises

Explain how you know that
.

Explain why
is not a real number but
is.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Learning Objectives

In this section, you will:

  • Express square roots of negative numbers as multiples of   ii.
  • Plot complex numbers on the complex plane.
  • Add and subtract complex numbers.
  • Multiply and divide complex numbers.

The study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. For example, we still have no solution to equations such as

x
2

+4=0

x
2

+4=0

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

Expressing Square Roots of Negative Numbers as Multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an . The imaginary number

i

i
is defined as the square root of negative 1.

−1

=i

−1

=i

So, using properties of radicals,

i
2

=

(

−1

)

2

=−1

i
2

=

(

−1

)

2

=−1

We can write the square root of any negative number as a multiple of

i.

i.
Consider the square root of –25.

−25

=

25⋅(−1)

         =

25

−1

         =5i

−25

=

25⋅(−1)

         =

25

−1

         =5i

We use

5i

5i
and not

−5i

−5i

because the principal root of

25

25
is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written

a+bi

a+bi
where

a

a
is the real part and

bi

bi
is the imaginary part. For example,

5+2i

5+2i
is a complex number. So, too, is

3+4
3
i
.

3+4
3
i
.

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

Imaginary and Complex Numbers

A is a number of the form

a+bi

a+bi
where

  • a

    a
    is the real part of the complex number.

  • bi

    bi
    is the imaginary part of the complex number.

If

b=0,

b=0,
then

a+bi

a+bi
is a real number. If

a=0

a=0
and

b

b
is not equal to 0, the complex number is called an imaginary number. An imaginary number is an even root of a negative number.

Given an imaginary number, express it in standard form.

  1. Write

    −a

    −a

    as

    a

    −1

    .

    a

    −1

    .

  2. Express

    −1

    −1

    as

    i.

    i.

  3. Write

    a

    ⋅i

    a

    ⋅i
    in simplest form.


Expressing an Imaginary Number in Standard Form

Express

−9

−9

in standard form.

Express

−24

−24

in standard form.

Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the , which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs

(a,b),

(a,b),
where

a

a
represents the coordinate for the horizontal axis and

b

b
represents the coordinate for the vertical axis.

Let’s consider the number The real part of the complex number is and the imaginary part is

3i.

3i.
We plot the ordered pair

(−2,3)

(−2,3)
to represent the complex number as shown in Figure 1.

Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in Figure 2.

The complex plane showing that the horizontal axis (in the real plane, the x-axis) is known as the real axis and the vertical axis (in the real plane, the y-axis) is known as the imaginary axis.

Given a complex number, represent its components on the complex plane.

  1. Determine the real part and the imaginary part of the complex number.
  2. Move along the horizontal axis to show the real part of the number.
  3. Move parallel to the vertical axis to show the imaginary part of the number.
  4. Plot the point.


Plotting a Complex Number on the Complex Plane

Plot the complex number

3−4i

3−4i
on the complex plane.

Plot the complex number

−4−i

−4−i
on the complex plane.

Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

Complex Numbers: Addition and Subtraction

Adding complex numbers:

(

a+bi

)+(

c+di

)=(

a+c

)+(

b+d

)i

(

a+bi

)+(

c+di

)=(

a+c

)+(

b+d

)i

Subtracting complex numbers:

(

a+bi

)−(

c+di

)=(

a−c

)+(

b−d

)i

(

a+bi

)−(

c+di

)=(

a−c

)+(

b−d

)i

Given two complex numbers, find the sum or difference.

  1. Identify the real and imaginary parts of each number.
  2. Add or subtract the real parts.
  3. Add or subtract the imaginary parts.


Adding Complex Numbers

Add

3−4i

3−4i
and

2+5i.

2+5i.

Subtract

2+5i

2+5i
from

3–4i.

3–4i.

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Number by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i.

Given a complex number and a real number, multiply to find the product.

  1. Use the distributive property.
  2. Simplify.


Multiplying a Complex Number by a Real Number

Find the product

4(2+5i).

4(2+5i).

Find the product

−4(2+6i).

−4(2+6i).

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

(

a+bi

)(

c+di

)=ac+adi+bci+bd
i
2

(

a+bi

)(

c+di

)=ac+adi+bci+bd
i
2

Because

i
2

=−1,

i
2

=−1,
we have

(

a+bi

)(

c+di

)=ac+adi+bci−bd

(

a+bi

)(

c+di

)=ac+adi+bci−bd

To simplify, we combine the real parts, and we combine the imaginary parts.

(

a+bi

)(

c+di

)=(

ac−bd

)+(

ad+bc

)i

(

a+bi

)(

c+di

)=(

ac−bd

)+(

ad+bc

)i

Given two complex numbers, multiply to find the product.

  1. Use the distributive property or the FOIL method.
  2. Simplify.


Multiplying a Complex Number by a Complex Number

Multiply

(

4+3i

)(2−5i).

(

4+3i

)(2−5i).

Multiply

(3−4i)(2+3i).

(3−4i)(2+3i).

Dividing Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of

a+bi

a+bi
is

a−bi.

a−bi.

Note that complex conjugates have a reciprocal relationship: The complex conjugate of

a+bi

a+bi
is

a−bi,

a−bi,
and the complex conjugate of

a−bi

a−bi
is

a+bi.

a+bi.
Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide

c+di

c+di
by

a+bi,

a+bi,
where neither

a

a
nor

b

b
equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

c+di

a+bi

 wherea≠0 andb≠0

c+di

a+bi

 wherea≠0 andb≠0

Multiply the numerator and denominator by the complex conjugate of the denominator.

(

c+di

)

(

a+bi

)

(

a−bi

)

(

a−bi

)

=

(

c+di

)(

a−bi

)

(

a+bi

)(

a−bi

)

(

c+di

)

(

a+bi

)

(

a−bi

)

(

a−bi

)

=

(

c+di

)(

a−bi

)

(

a+bi

)(

a−bi

)

Apply the distributive property.

=

ca−cbi+adi−bd
i
2

a
2

−abi+abi−
b
2

i
2

=

ca−cbi+adi−bd
i
2

a
2

−abi+abi−
b
2

i
2

Simplify, remembering that

i
2

=−1.

i
2

=−1.

=

ca−cbi+adi−bd(−1)

a
2

−abi+abi−
b
2

(−1)

=

(ca+bd)+(ad−cb)i

a
2

+
b
2

=

ca−cbi+adi−bd(−1)

a
2

−abi+abi−
b
2

(−1)

=

(ca+bd)+(ad−cb)i

a
2

+
b
2

The Complex Conjugate

The of a complex number

a+bi

a+bi
is

a−bi.

a−bi.
It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

  • When a complex number is multiplied by its complex conjugate, the result is a real number.
  • When a complex number is added to its complex conjugate, the result is a real number.


Finding Complex Conjugates

Find the complex conjugate of each number.

  1. 2+i
    5

    2+i
    5


  2. 1
    2

    i


    1
    2

    i

Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by

i.

i.

Given two complex numbers, divide one by the other.

  1. Write the division problem as a fraction.
  2. Determine the complex conjugate of the denominator.
  3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
  4. Simplify.


Dividing Complex Numbers

Divide

(

2+5i

)

(

2+5i

)
by

(

4−i

).

(

4−i

).


Substituting a Complex Number into a Polynomial Function

Let

f(x)=
x
2

−5x+2.

f(x)=
x
2

−5x+2.
Evaluate

f(

3+i

).

f(

3+i

).

We write

f(3+i)=−5+i.

f(3+i)=−5+i.
Notice that the input is

3+i

3+i
and the output is

−5+i.

−5+i.

Let

f(x)=2
x
2

−3x.

f(x)=2
x
2

−3x.
Evaluate

f(

8−i

).

f(

8−i

).


Substituting an Imaginary Number in a Rational Function

Let

f(
x
)=

2+x

x+3

.

f(
x
)=

2+x

x+3

.
Evaluate

f(

10i

).

f(

10i

).

Let

f(x)=

x+1

x−4

.

f(x)=

x+1

x−4

.
Evaluate

f(

−i

).

f(

−i

).

Simplifying Powers of i

The powers of

i

i
are cyclic. Let’s look at what happens when we raise

i

i
to increasing powers.

i
1

=i

i
2

=−1

i
3

=
i
2

⋅i=−1⋅i=−i

i
4

=
i
3

⋅i=−i⋅i=−
i
2

=−(−1)=1

i
5

=
i
4

⋅i=1⋅i=i

i
1

=i

i
2

=−1

i
3

=
i
2

⋅i=−1⋅i=−i

i
4

=
i
3

⋅i=−i⋅i=−
i
2

=−(−1)=1

i
5

=
i
4

⋅i=1⋅i=i

We can see that when we get to the fifth power of

i,

i,
it is equal to the first power. As we continue to multiply

i

i
by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of

i.

i.

i
6

=
i
5

⋅i=i⋅i=
i
2

=−1

i
7

=
i
6

⋅i=
i
2

⋅i=
i
3

=−i

i
8

=
i
7

⋅i=
i
3

⋅i=
i
4

=1

i
9

=
i
8

⋅i=
i
4

⋅i=
i
5

=i

i
6

=
i
5

⋅i=i⋅i=
i
2

=−1

i
7

=
i
6

⋅i=
i
2

⋅i=
i
3

=−i

i
8

=
i
7

⋅i=
i
3

⋅i=
i
4

=1

i
9

=
i
8

⋅i=
i
4

⋅i=
i
5

=i


Simplifying Powers of

i

i

Evaluate

i

35

.

i

35

.

Can we write

i

35

i

35

in other helpful ways?

As we saw in Example 10, we reduced

i

35

i

35

to

i
3

i
3

by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of

i

35

i

35

may be more useful. Table 1 shows some other possible factorizations.

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

3.1 Section Exercises

Verbal

Explain how to add complex numbers.

What is the basic principle in multiplication of complex numbers?

Give an example to show the product of two imaginary numbers is not always imaginary.

What is a characteristic of the plot of a real number in the complex plane?

Algebraic

Iff(x)=
x
2

+x−4,

Iff(x)=
x
2

+x−4,
evaluate

f(2i).

f(2i).

Iff(x)=
x
3

−2,

Iff(x)=
x
3

−2,
evaluate

f(i).

f(i).

Iff(x)=
x
2

+3x+5,

Iff(x)=
x
2

+3x+5,
evaluate

f(2+i).

f(2+i).

Iff(x)=2
x
2

+x−3,

Iff(x)=2
x
2

+x−3,
evaluate

f(2−3i).

f(2−3i).

Iff(x)=

x+1

2−x

,

Iff(x)=

x+1

2−x

,
evaluate

f(5i).

f(5i).

Iff(x)=

1+2x

x+3

,

Iff(x)=

1+2x

x+3

,
evaluate

f(4i).

f(4i).

Graphical

Graph of a parabola not intersecting the real axis.

Numeric

(

3+2i

)+(5−3i)

(

3+2i

)+(5−3i)

(

−2−4i

)+(

1+6i

)

(

−2−4i

)+(

1+6i

)

(

−5+3i

)−(6−i)

(

−5+3i

)−(6−i)

(

2−3i

)−(3+2i)

(

2−3i

)−(3+2i)

(−4+4i)−(−6+9i)

(−4+4i)−(−6+9i)

(

2+3i

)(4i)

(

2+3i

)(4i)

(

5−2i

)(3i)

(

5−2i

)(3i)

(

6−2i

)(5)

(

6−2i

)(5)

(

−2+4i

)(
8
)

(

−2+4i

)(
8
)

(

2+3i

)(4−i)

(

2+3i

)(4−i)

(

−1+2i

)(−2+3i)

(

−1+2i

)(−2+3i)

(

4−2i

)(4+2i)

(

4−2i

)(4+2i)

(

3+4i

)(

3−4i

)

(

3+4i

)(

3−4i

)

−4

−4

−25

−4

−4

−25

Technology

Evaluate

(1+i)

k

(1+i)

k

for

k=4, 8, and 12.

k=4, 8, and 12.
Predict the value if

k=16.

k=16.

Evaluate

(1−i)

k

(1−i)

k

for

k=2, 6, and 10.

k=2, 6, and 10.
Predict the value if

k=14.

k=14.

Evaluate

(1+i)k

(1−i)
k

(1+i)k

(1−i)
k
for

k=4, 8, and 12

k=4, 8, and 12
. Predict the value for

k=16.

k=16.

Show that a solution of

x
6

+1=0

x
6

+1=0
is

3

2

+
1
2

i.

3

2

+
1
2

i.

Show that a solution of

x
8

−1=0

x
8

−1=0
is

2

2

+

2

2

i.

2

2

+

2

2

i.

Extensions

1
i

+
4

i
3

1
i

+
4

i
3

1

i

11


1

i

21

1

i

11


1

i

21

i
7

(

1+
i
2

)

i
7

(

1+
i
2

)

i

−3

+5
i
7

i

−3

+5
i
7

(

2+i

)(

4−2i

)

(1+i)

(

2+i

)(

4−2i

)

(1+i)

(

1+3i

)(

2−4i

)

(1+2i)

(

1+3i

)(

2−4i

)

(1+2i)

(

3+i

)

2

(

1+2i

)

2

(

3+i

)

2

(

1+2i

)

2

3+2i

2+i

+(

4+3i

)

3+2i

2+i

+(

4+3i

)

4+i

i

+

3−4i

1−i

4+i

i

+

3−4i

1−i

3+2i

1+2i

2−3i

3+i

3+2i

1+2i

2−3i

3+i

Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with roots
  • Estimate and approximate roots
  • Simplify variable expressions with roots


Before you get started, take this readiness quiz.

Simplify:

If you missed this problem, review Example 2.21.


Round to the nearest hundredth.

If you missed this problem, review Example 1.34.


Simplify:

If you missed this problem, review Example 5.12.

Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write and read it ‘n squared’. This number is called the of n, and n is called the . For example,

132is read “13 squared”169 is called thesquareof 13, since132=16913 is asquare rootof 169132is read “13 squared”169 is called thesquareof 13, since132=16913 is asquare rootof 169

Square and Square Root of a number

Ifn2=m,thenmis thesquareofn.Ifn2=m,thenmis thesquareofn.

Ifn2=m,thennis asquare rootofm.Ifn2=m,thennis asquare rootofm.

Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are .

We also use the radical sign for the square root of zero. Because Notice that zero has only one square root.

Square Root Notation

mis read “the square root ofm”.Ifn2=m,thenn=m,forn≥0.mis read “the square root ofm”.Ifn2=m,thenn=m,forn≥0.

The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled “radical sign”. The variable m is labeled “radicand”.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example,


Simplify:


Simplify:


Simplify:

Can we simplify Is there a number whose square is

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to The square root of a negative number is not a real number.


Simplify:


Simplify:


Simplify:

So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

We write:We say:n2nsquaredn3ncubedn4nto the fourth powern5nto the fifth powerWe write:We say:n2nsquaredn3ncubedn4nto the fourth powern5nto the fifth power

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from −5 to 5. See Figure 8.2.

The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125.

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this.

The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label “Even power Positive result”. Arrows point to the first third and fifth columns with the label “Odd power Negative result”.

We will now extend the square root definition to higher roots.

nth Root of a Number

Ifbn=a,thenbis annthroot ofa.The principalnthroot ofais writtenan.nis called theindexof the radical.Ifbn=a,thenbis annthroot ofa.The principalnthroot ofais writtenan.nis called theindexof the radical.

Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for

We can refer to Figure 8.2 to help find higher roots.

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of

a
n

a
n


When n is an even number and

  • a≥0,a≥0, then anan is a real number.
  • a<0,a<0, then anan is not a real number.

When n is an odd number, is a real number for all values of a.

We will apply these properties in the next two examples.


Simplify:


Simplify:


Simplify:

In this example be alert for the negative signs as well as even and odd powers.


Simplify:


Simplify:


Simplify:

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the what is the value of or In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for numbers closest to the radicand. To find an estimate of we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.

The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Square Root”. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Cube Root”. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.

Similarly, to estimate we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.


Estimate each root between two consecutive whole numbers:


Estimate each root between two consecutive whole numbers:


Estimate each root between two consecutive whole numbers:

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the key on your calculator. To find a cube root, or any root with higher index, you will use the key.

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

5≈2.236067978rounded to two decimal places is5≈2.24934≈3.105422799rounded to two decimal places is934≈3.115≈2.236067978rounded to two decimal places is5≈2.24934≈3.105422799rounded to two decimal places is934≈3.11

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.


Round to two decimal places:


Round to two decimal places:


Round to two decimal places:

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

Three equivalent expressions are written: the cube root of 4 cubed, the cube root of 64, and 4. There are arrows pointing to the 4 that is cubed in the first expression and the 4 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the cube root of the quantity negative 4 in parentheses cubed, the cube root of negative 64, and negative 4. The negative 4 in the first expression and the negative 4 in the last expression are labeled as being the “same”.

But what about an even root? We want the principal root, so

Three equivalent expressions are written: the fourth root of the quantity 5 to the fourth power in parentheses, the fourth root of 625, and 5. There are arrows pointing to the 5 in the first expression and the 5 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the fourth root of the quantity negative 5 in parentheses to the fourth power in parentheses, the fourth root of 625, and 5. The negative 5 in the first expression and the 5 in the last expression are labeled as being the “different”.

Simplifying Odd and Even Roots

For any integer

We must use the signs when we take an even root of an expression with a variable in the radical.


Simplify:


Simplify:


Simplify:

What about square roots of higher powers of variables? The Power Property of Exponents says So if we square am, the exponent will become 2m.

Looking now at the square root,

We apply this concept in the next example.


Simplify:


Simplify:


Simplify:

The next example uses the same idea for highter roots.


Simplify:


Simplify:


Simplify:

But notice and no absolute value sign is needed as u4 is always positive.


Simplify:


Simplify:


Simplify:

This example just takes the idea farther as it has roots of higher index.


Simplify:


Simplify:


Simplify:

The next examples have two variables.


Simplify:


Simplify:


Simplify:

Section 8.1 Exercises

Practice Makes Perfect

Simplify Expressions with Roots

Estimate and Approximate Roots

Simplify Variable Expressions with Roots

Writing Exercises

Why is there no real number equal to

What is the difference between and

Explain what is meant by the nth root of a number.

Explain the difference of finding the nth root of a number when the index is even compared to when the index is odd.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with roots.”, “estimate and approximate roots”, and “simplify variable expressions with roots”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

If most of your checks were:

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