- Squaring Removes Any Negative
- Even Exponents of Negative Numbers
- Example: What is (−1)97 ?
- Example: What is (−2)6 ?
- Roots of Negative Numbers
- Example: What is the value of x here: x2 = −1
- How Many of The Roots are Positive?
- Complex Roots
- Improving the Number of Positive Roots
- How Many of The Roots are Negative?
- But remember to reduce it because there may be Complex Roots!
- Total Number of Roots
- What we Know
- Must Have a Constant Term
- Example: 2x4 + 3x2 − 4x
- nth Root
- The nth Root
- The nth Root Symbol
- Using it
- Why «Root» … ?
- Properties
- Multiplication and Division
- Example:
- Addition and Subtraction
- Exponents vs Roots
- Example:
- nth Root of a-to-the-nth-Power
- nth Root of a-to-the-mth-Power
- Example:
- Example:
- Learning Objectives
- Simplify Expressions with Higher Roots
- nth Root of a Number
- Properties of a n a n
- Simplifying Odd and Even Roots
- Use the Product Property to Simplify Expressions with Higher Roots
- Simplified nth Root
- Product Property of nth Roots
- Use the Quotient Property to Simplify Expressions with Higher Roots
- Quotient Property of nth Roots
- Add and Subtract Higher Roots
- Section 9.7 Exercises
- Practice Makes Perfect
- Everyday Math
- Writing Exercises
- Self Check
- Learning Objectives
- Expressing Square Roots of Negative Numbers as Multiples of i
- Imaginary and Complex Numbers
- Expressing an Imaginary Number in Standard Form
- Plotting a Complex Number on the Complex Plane
- Plotting a Complex Number on the Complex Plane
- Adding and Subtracting Complex Numbers
- Complex Numbers: Addition and Subtraction
- Adding Complex Numbers
- Multiplying Complex Numbers
- Multiplying a Complex Number by a Real Number
- Multiplying a Complex Number by a Real Number
- Multiplying Complex Numbers Together
- Multiplying a Complex Number by a Complex Number
- Dividing Complex Numbers
- The Complex Conjugate
- Finding Complex Conjugates
- Dividing Complex Numbers
- Substituting a Complex Number into a Polynomial Function
- Substituting an Imaginary Number in a Rational Function
- Simplifying Powers of i
- Simplifying Powers of i i
- 3.1 Section Exercises
- Verbal
- Algebraic
- Graphical
- Numeric
- Technology
- Extensions
- Learning Objectives
- Simplify Expressions with Roots
- Square and Square Root of a number
- Square Root Notation
- nth Root of a Number
- Properties of a n a n
- Estimate and Approximate Roots
- Simplify Variable Expressions with Roots
- Simplifying Odd and Even Roots
- Section 8.1 Exercises
- Practice Makes Perfect
- Writing Exercises
- 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:
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:
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):
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 Roots always come in 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?
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:
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).
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:
There is just one sign change,
So there is 1 positive root
And the negative case (after flipping signs of odd-valued exponents):
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
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!
+
a − b
−
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
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 7√2d — 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.
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.
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.
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
n
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.
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
.
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.
- Write
−a
−a
as
a
−1
.
a
−1
.
- Express
−1
−1
as
i.
i.
- 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.
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in Figure 2.
Given a complex number, represent its components on the complex plane.
- Determine the real part and the imaginary part of the complex number.
- Move along the horizontal axis to show the real part of the number.
- Move parallel to the vertical axis to show the imaginary part of the number.
- 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.
- Identify the real and imaginary parts of each number.
- Add or subtract the real parts.
- 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,
Given a complex number and a real number, multiply to find the product.
- Use the distributive property.
- 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.
- Use the distributive property or the FOIL method.
- 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.
2+i
52+i
5−
1
2i
−
1
2i
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.
- Write the division problem as a fraction.
- Determine the complex conjugate of the denominator.
- Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
- 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
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.
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.
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.
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
n
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.
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,
But what about an even root? We want the principal root, so
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.
If most of your checks were: