Basic Math & Pre-Algebra

General Root Calculator

In mathematics, the general root, or the nth root of a number a is another number b that when multiplied by itself n times, equals a. In equation format:

n√ = b
bn = a

Estimating a Root

Some common roots include the square root, where n = 2, and the cubed root, where n = 3. Calculating square roots and nth roots is fairly intensive. It requires estimation and trial and error. There exist more precise and efficient ways to calculate square roots, but below is a method that does not require a significant understanding of more complicated math concepts. To calculate √:

Estimating an nth Root

It should then be clear that computing any further will result in a number that would round to 1.403, making 1.403 the final estimate to 3 decimal places.

In this article we will look at the cube root of the number 2 and show you how to calculate it and determine if it is a perfect cube or not.

What is a cube root?

The cube root of 2 is another number that when multiplied by itself twice, would be exactly equal to 2.

We would normally express this problem in mathematical form by using the cube root symbol with the number alongside it:

You might notice that this symbol is very similar to the square root symbol. The only difference is the number, called the index, above the symbol, which lets us know this a cube root calculation.

To recap: the √ symbol itself is called the radical symbol, the number above it is called the index, and the actual number, 2, is called the radicand.

The basic formula we use to calculate a cube number from the result is a number which we’ll call n in this example, that when multiplied by itself twice, is equal to 2:

n × n × n = 2

When we use ∛2, the actual math calculation we perform is 2⅓. This means you can use that formula in Excel, Google Sheets, or Mac Numbers to calculate the cube root:

You should now be able to confirm that the cube root of 2 is:

∛2 ≈ 1.2599210498949

Is 2 a perfect cube number?

Whenever the cube root of a given number is a whole number without any fractional or decimal part, we call this a perfect cube.

These are important for a lot of different math functions and can be used in simpler settings like carpenty and room design all the way through to more advanced calculations in physics and astronomy.

For the number 2 we have already calculated the answer of 1.2599210498949 using a scientific calculator and since this is not a whole number, we also know that 2 is not a perfect cube number.

We have a list of perfect cubes article which covers the first 500 perfect cube numbers for you.

Is the cube root of 2 a rational or irrational number?

This is a commonly asked question and the answer to whether the cube root of 2 is rational or irrational is always the same as if it is a perfect cube.

Remember that a rational number can be written as a fraction, while irrational numbers can’t.

So, if a number is a perfect cube, then it’s a rational number, but if it is not a perfect cube then it is an irrational number.

We just confirmed that 2 is not a rational number then, because we know it is not a perfect cube number.

Rounding the cube root of 2

There might be occasions when you need to round the result of a cube root calculation down to a specific number of decimal places. Below, we show an example of a few of these to specific decimal places that show up in exam questions:

10th: ∛2 = 1.3

100th: ∛2 = 1.26

1000th: ∛2 = 1.260

Simplifying the cube root of 2

Sometimes, the radicand can be simplified and made smaller. If that is possible, we call it the cube root of 2 in its simplest form.

The cube root of 2 cannot be simplified down any further in this example, so there are no further calculations to be made.

Practice perfect cube and cube roots using examples

Still here and interested in perfect cubes? You must really love yourself some math!

Take a look at the random calculations in the sidebar on the right hand side of this page and click through to some of the cube root calculations.

How many of them are perfect cubes? You can also try to think of and calculate some perfect cubes yourself without looking at the list above to test yourself.

Calculate a cube root problem

If you have an example you want to try and see if it is a perfect cube, enter it in the form below and click calculate to find out if it is.

Enter your number in box A below and click «Calculate» to work out the cube root of the given number.

Use this cube root calculator to easily calculate the cube root of a given number.

The cube root of a number answers the question «what number can I multiply by itself twice to get this number?». It is the reverse of the exponentiation operation with an exponent of 3, so if r3 = x, then we say that «r is the cube root of x». Finding the root of a number has a special notation called the radical symbol: √ — which is used as it is when it comes to square roots, but gets an index number indicating the root — 3 in the case of the cube root, so it looks like so: ∛. Usually the radical spans over the entire equation for which the root is to be found. It is called a «cube» root since multiplying a number by itself twice is how one finds the volume of a cube.

When interpreting the output of the calculator it might help to know that in geometrical terms, the cube root function maps the volume of a cube onto its side length.

Properties and practical application of the cube root

Cubic roots are useful when one needs to divide an angle in three, that is — to find an angle whose measure is one third of a given angle. This operation is called angle trisection. Cube roots are used when you need to find the edge of a cube whose volume is twice that of a cube with a given edge.

Plotting the results from the cube root function, as calculated using this calculator, on a graph reveals that it has the shape of half a parabola.

Commonly used cube roots

Table of commonly encountered cube roots:

The calculations were performed using this cube root calculator.

Does the calculator support fractions?

Yes, simply enter the fraction as a decimal floating point number and you will get the corresponding cube root. For example, to compute the cube root of 1/2 simply enter 0.5 in the input field and you will get 0.7937 as ouput. If you are having trouble converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

Use this online calculator to easily calculate the square root of a given number, including fractions. Quick and easy square root finder.

What is a square root?

The square root of a number answers the question «what number can I multiply by itself to get this number?». It is the reverse of the exponentiation operation with an exponent of 2, so if r2 = x, then we say that «r is the root of x». Finding the root of a number has a special notation called the radical symbol: √. Usually the radical spans over the entire equation for which the root is to be found. It is called a «square» root since multiplying a number by itself is called «squaring» as it is how one finds the area of a square.

For every positive number there are two square roots — one positive and one negative. For example, the square root of 4 is 2, but also -2, since -2 x -2 = 4. The negative root is always equal in value to the positive one, but opposite in sign. You can see examples in the table of common roots below. Most often when talking about «the root of» some number, people refer to the Principal Square Root which is always the positive root. This is the number our square root calculator outputs as well.

In geometrical terms, the square root function maps the area of a square onto its side length. The function √x is continuous for all nonnegative x and differentiable for all positive x.

How calculate a square root

For example, to find the square root of 30 with a precision of three numbers after the decimal point:

Step 1: a = 30 is between 25 and 36, which have roots of 5 and 6 respectively. Let us start with b = 5.5.
Step 2: e = a / b = 30 / 5.5 = 5.45(45). Since b is not equal to e (5.500 ≠ 5.454), continue calculation.
Step 3: b1 = (5.45 + 5.5) / 2 = 5.47727(27)
Step 4: e = 30 / 5.47727 = 5.477178. Since b1 = e = 5.477 within three position after the decimal point, stop the square root-finding algorithm with a result of √30 = 5.47727(27).

Checking the outcome against the square root calculator output of 5.477226 reveals that the algorithm resulted in a correct solution. While the above process can be fairly tedious especially with larger roots, but will help you find the square root of any number with the desired decimal precision.

Properties and practical application of square roots

Square roots appear frequently in mathematics, geometry and physics. For example, many physical forces measured in quantities or intensities diminish inversely proportional to the square root of the distance. So, gravity between two objects 4 meters apart will be 1/√4 relative to their gravity at 0 meters. The same is true for radar energy waves, radio waves, light and magnetic radiation in general, and sound waves in gases. It is usually referred to as the «inverse-square law».

The square root is key in probability theory and statistics where it defines the fundamental concept of standard deviation.

Plotting the results from the square root function, as calculated using this square root calculator, on a graph reveals that it has the shape of half a parabola.

Commonly used square roots

Table of commonly encountered square roots:

The calculations were performed using this calculator.

Yes, simply enter the fraction as a decimal number (use dot as a separator) and you will get the corresponding root. For example, to compute the square root of 1/4 simply enter 0.25 in the number field, press «Calculate» and you will get 0.50 as ouput. If you are having difficulty converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

I was required to find the cube-root of \$a^3\$ as I had to use \$a\$ to find another quantity which was actually my answer.

I know that the cube-root of \$a^3\$ here can be found by using logarithm.

Now, taking antilog both sides, this becomes : \$\$a = antilog(-7.481) \$\$

Now, as you all might have noticed that this is a lot longer method and this has larger probability of a student doing a mistake in the calculations etc.

Is there a shorter or any proper method to find cube-root of a number like stated above?

Square root of number \$-1\$ defined as \$i\$

That’s the root of the problem: that people are taught that \$i\$ is «defined to be» something, and not that it comes naturally from the geometry and arithmetics of the problem. Because if something is «defined to be» this and that, it is automatically made non-debatable and non-explainable.

If numbers were discovered in the correct order, and in the correct way, then you would see why \$i\$ is what it is, because it naturally comes out of arithmetics. Here’s how you can discover \$i\$ yourself, already with its proper geometric interpretation:

Now we have everything we need to discover \$i\$ (and \$-1\$ along the way, too!):

\$\$1:x = x:1\$\$

Or in words: You do something with your unit (\$1\$) to get \$x\$. What do you need to to with that \$x\$ to get your unit (\$1\$) back?

The obvious answer is: Just do nothing. Let \$x = 1\$ (the number \$1\$ is the number which doesn’t change any other number when multiplied with it; it is a multiplicative identity). Let’s check: Substitute \$1\$ as \$x\$, and you’ll get:

\$\$1:(1) = (1):1\$\$

But there’s one more answer to this puzzle: You can turn your unit by \$180°\$, making it to operate in the opposite direction. That is, the other answer is \$x = -1\$, which is the opposite of \$1\$. Let’s check:

\$\$1:(-1) = (-1):1\$\$

The proportion above says that whatever you do with your unit (\$1\$) to get \$-1\$ (which is turning it by \$180°\$), you do the same with the \$-1\$ obtained this way (that is, also rotate it by \$180°\$) to get your standard unit \$1\$ back. And this is true: Turning your unit twice by \$180°\$ will turn it by \$360°\$, which is the same as turning it by \$0°\$, or not turning it at all. Doing an operation twice is the same as raising to the second power, so what it says is that \$1cdot1 = 1^2 = 1\$ and also \$(-1)cdot(-1) = (-1)^2 = 1\$. Both \$1\$ and \$-1\$ are then square roots of unity. You can also see that if you didn’t know negative numbers, you could discover them with geometric progressions: \$1\$ is its own geometric mean, but also \$-1\$ is a geometric mean between \$1\$ and \$1\$ (or half-way in between them in multiplication).

\$\$1 : x = x : -1\$\$

To turn \$1\$ into \$-1\$, you need to literally turn it by \$180°\$. But here you need to somehow make half of that turn, twice. And what is the half-turn of \$180°\$? Of course, it is \$90°\$! 🙂 So your \$x\$, which is a square root of \$-1\$ (that is, an operation you need to apply twice on your unit to turn it into \$-1\$), turns out (huh :P) to be at the \$90°\$ to your standard unit. You can then call it an imaginary unit as it is commonly being called. Let’s check:

But remember that there are always two answers to such geometric progressions. The other one is, in this case, the opposite of \$i\$, that is, \$-1\$. Let’s check:

which is also true, and you can also verify it geometrically: multiplying by \$-i\$ is rotating by \$270°\$ (or, as I prefer to see it: by \$-90°\$, that is, by \$90°\$ in the opposite direction).

So you can conclude that:

\$\$i^2 = -1\$\$, or:

then what is the square root of complex number \$i\$?

To find out what are the square roots of \$i\$, you need to do exactly the same thing you did up to this point: You need to find a geometric mean between \$1\$ and \$i\$. Or, in symbols:

\$\$1 : x = x : i\$\$

Of course this is not the only answer! Can you find the others? 😉 I’ll leave it as an exercise for you. Now, when you know how it works, I’m sure you’ll be able to find them.

I would say it should be \$j\$ as logic suggests

WHAT logic suggests that?

but it’s not defined in quaternion theory in that way, am I wrong?

Yes, you are wrong. With quaternions you have:

\$\$i^2 = j^2 = k^2 = i cdot j cdot k = -1\$\$

so the quaternion unit \$j\$ is a different animal than your \$j\$: your \$j^2 = i\$, and the quaternion \$j^2 = -1\$.

my question is rather related to nomenclature of definition, while square root of -1 defined as i, why not j defined as square root of i and k square root of j and if those numbers have deeper meanings and usage as in quaternions theory

Blurting out whatever definitions you like is always easy. But the real art is to make definitions which make sense and understand what meaning hides behind them.

Whenever you come across the word cube root the two words that come into our mind are cube and roots of a tree. Actually the concept is a bit similar in this sense, root actually refers to the primary source of origin. So we should think about what number you should refer to. That cube will give you a particular number that you are looking for.

Cube root of a number x is a number y only when  y×y×y= x. All the nonzero real numbers have one real cube root and along with it a pair of complex conjugate cube roots, and all non zero complex numbers have three different complex roots that are cube roots.

The cube root of any number in brief can be defined as the factor that we multiply by itself three times to get the particular numbers. Remember that the cube root of a number is exactly opposite of the cubing of a number.

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512 etc.

To find the volume of the cube, we have volume = side3, but if we want to find the side of a cube we have to take the cube root of the volume. Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbol is 3√.

Let’s suppose we need to find the value of cube root of 2 is a value that is obtained by multiplying that number three times. It is expressed in the form of ‘3√2’. The meaning of cube root is basically the root of a number that is generated by taking the cube of another number. Hence, if the value of 3√2=x, then x =2 and we need to find the value of x.

We can define the cube root of a number as a special value that, when used in a multiplication exactly three times, gives us that number.For example, 3 × 3 × 3 equals 27, so the cube root of 27 is 3.

The Cube Root Symbol

The special symbol given below signifies the «cube root», it is known to be the «radical» symbol (the symbol can be used for square roots) and with a little three to mean cube root.

You can use it like this, the cube root of 27 is : 3√27=3 (we say «the cube root of 27 equals to 3»)

Find the Cube of Fraction

You can find the cube of a fraction the same way as you are finding the cube of a number. You just need to multiply the fraction three times. Just take the example of ⅔. You can get the cube of this fractional number by first multiplying the number in the numerator 3 times that is here you can first multiply the number 2 three times, 2×2×2 and as a result you will get 8 that is the Cube of the number 2 which is in your numerator.

Now move towards the denominator, here in your denominator there is 3, you can get its cube by multiplying it 3 times, that is 3×3×3 hereafter cubing the number 3 you are going to get 27 as a result, which is the Cube of your number 3, registered here as the denominator. So as a result you are going to get 8/27. This 8/27 is the Cube of your number ⅔ which you can get either by doing the multiplication of these numbers separately or you can do the multiplicity of the whole fraction three times, that is ⅔ × ⅔ × ⅔. Giving you the result of 8/27.

You Can Also Cube Negative Numbers

Have a look at this:When we cube +5 we generally get +125:  +5 × +5 × +5 = +125

When we cube −5 we get the number −125:  −5 × −5 × −5 = −125. So the cube root of the number −125 is equal to −5

Cube Roots (For Integer Results 1 through 10)

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself.

Let’s see for example,

23 =8, or the cube root of the number 8 is 2

33 = 27, or the cube root of the number 27 is 3

43 = 64, or the cube root of 64 is 4

53 = 125, or the cube root of 125 is 5

The symbol of the cube root is a

Thus, the cube root of 125 is represented as 3√125=5 and that of 27 can be represented as 3√27 equals 3 and so on.

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number can be defined as the inverse operation of cubing a number.

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If the cube of a number 63 = 216

Then the cube root of ∛216 is equal to 6.

Cube root of any largest number can be easily found in four ways:

A Few Properties of Cube Root

A perfect cube of  an integer is that integer which is actually equal to some other integer raised to the third power. We refer to raising the number to the third power as cubing the number.

Perfect Cube :1 8 27 64 125 216 343 512 729 1000 .

What is a Cube Root Calculator?

Cube root calculator is a tool that will help you to find the cube root of a particular number. This calculator is free. By using the cube root calculator you can find the cube root of a number free without giving any charge.

How can you use a Cube Root Calculator?

What is the Cube Root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

Now we are getting closer, but slowly at this point, we can use a calculator and it says:

What is the Cube Root of 1728?

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself. The cube root is the inverse operation of cubing a number. The cube root symbol is , it is the “radical” symbol (used for square roots) with a little three to mean cube root.

Perfect Cube Root

A cube is a number that will result in a number multiplied by itself three times for example if y is a perfect cube root of z then y = z. If the number 5 is multiplied 3 times it results in 125 so 125 is a perfect cube root of 5.

Perfect cube root of any number = number x number x number.

A perfect cube root will always give an integer answer but a non-perfect cube root will not give an integer answer. Perfect cube root will be free from decimals and fractions.

Cube root list 1 to 100 will help students to solve the cube root problem easily, accurately, and with speed.

Cube Root of 1 to 100

Finding the Cube Root of a Perfect Cube

Recall that a perfect cube is a number that is the result of multiplying a number with itself  3 times.

We can think of cube roots in the same context that we view square roots. When we take the square root of a perfect square, we are searching for the number that, when multiplied by itself two times, results in the perfect square. Similarly, when we are finding the cube root of the perfect cube, we are searching for the number that when multiplied by itself three times, results in the perfect cube.

Let’s solve an example.

Solution: We know,

Did You Know?

This is the cube root list of numbers from 1 to 100. Concentrate on how the cube roots of different numbers are calculated. Focus on the features of cube roots along with examples and develop your concepts well.

Square Root and Cube Root are the important concepts used in Mathematics. As we have seen square numbers can be represented by a two-dimensional square, a cube number can be represented by a three-dimensional cube.

We calculate the area of a square as side x side i.e side, but if we have to calculate the side of a square we need to take the square root of the area. Thus we can say that the square root is the inverse operation of squaring a number.

On this page, we will be learning square roots and cube roots definition, square root, and cube root symbol and square root and cube root example.

Square Roots and Cube Roots Symbol

The square root of a number x is that number which when multiplied by itself gives the number x itself. The number x is a perfect square.

For Example, 2=4, or the square root of 4 is 2

=9, or the square root of 9 is 3

= 16, or the square root of 16 is 4

For Example, 2=8, or the cube root of 8 is 2

= 27, or the cube root of 27 is 3

Properties of Square Root

This method will help you to find the square roots and cube root of a given number but if the square root and cube root of the first 10 numbers are memorized it will help you solve your problems more quickly. Here is the  square root and cube root table format, which will help you to memorize these square roots and cube roots.

Square Root and Cube Root of 1 to 15

Square root list and cube root list of 1 to 15 will help you to solve the most time consuming long equations within no time. This square root and cube root table will be beneficial to you at every step.

Make use of this square root list and cube root list and solve the below square root and cube root example.

Example 1: Find the cube root of 2744

By Prime Factorisation method

Step 1: First we take the prime factors of a given number

2744 = 2 x 7 x 2 x 2 x 7 x 7

Step 2: Form groups of three similar factors

= 2 x 2 x 2 x 7 x 7 x 7

Step 3: Take out one factor from each group and multiply.

Example 2: Find the cube root of 1728 by long division method

= 2 x 2 x 3

Some more square root and cube root examples to solve.

Introduction To Cube Root

The word root means the source, so by the literal definition, cube root means the cube of any number must be taken to get the original number. The product of a number multiplied thrice (three times) by itself is the  of that specific number. A cube root is the inverse of the cube of any number, so if we take the cube root of a number, then the answer will always be the original number.

Figure 1 – Cube Root of 1

Let us take a number 2 to understand the concepts of cube roots. We know that when this number is multiplied three times, we get 8 as 2 × 2 × 2 = 8. So the number 8 is the cube of 2, which can be written as:

23 = 8

Now to take the cube root of this number, we will use the inverse method. As discussed above cube root is a factor of the number itself:

The cube root of 8 is, therefore, 2.

To solve the mathematical questions, we use the cube root formula to ease us. We can do this by finding the prime factorization of the number. After that, we take the cube root to the answer. Let us take the example of 64:

64 = 4 × 4 × 4

Now applying the cube root formula:

Thus, 4 is the cube root of 64.

When we multiply 3 same integers with each other then the resulting cube is known as the perfect cube and similarly, the square root will be a whole number integer. Such numbers are known as cube numbers.

Figure 2 – Perfect Cube of A

Let us take the example of the number 27:

33 = 3 × 3 × 3 = 27

Now taking the cube root:

Hence 27 is a perfect cube.

Not every number is a perfect cube if we take the example of the number 34 then we get:

This is because it does not satisfy the condition in which the cube root must be a whole number integer.

Cube Root of Negative Number

Let us consider a number x which is a negative number so to take its cube root we will rewrite it again while separating the negative sign from it:

After separating the negative sign we do prime factorization on x and list all possible factors. We will then make groups of three and multiply them together so that we can get the cube root of x.

Note that the cube root of a negative number will always be a negative number and similarly the cube root of a positive number is also positive.

Examples of Cube Root

Hailey is a part of a gaming club where you have to collect badges to level up. The highest record in this club is 216 badges. Find the number of Hailey’s badge if her level is the cube root of the highest record holder. Then tell how many more badges she needs to earn to become the record holder.

From the statement given above, we can say that the highest record is 216 so:

So this tells that Hailey is on level 6 at the moment and to beat the record she needs:

216 – 6 = 210

210 badges are required for Hailey to reach the record but to break it she needs 1 more badge:

210 + 1 = 211 badges

Todd wants to find the cube root of 2,744 for his math assignment. Help him find it.

We will solve this by using the prime factorization method so:

Figure 3 – Factor Tree Representing Prime Factorization of 2744

2744 = 2 × 2 × 2 × 7 × 7 × 7

Now we will group them so that we have 3 numbers in each pair:

2744 = (2 × 2 × 2) × (7 × 7 × 7)

2744 = 23 × 73

= 2 × 7

So the Cube root of 2744 is the number 14.

First, we use the prime factorization method:

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Now we will group them together:

512 = (2 ×2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

512 = 23 × 23 × 23

Hence 512 is a Perfect Cube.

48 = 2 × 2 × 2 × 2 × 3

48 = (2 × 2 × 2) × (2) × (3)

48 = 23 × 2 × 3

Hence 48 is not a Perfect Cube.

Images/mathematical drawings are created with GeoGebra.

Cube Number Definition < Glossary Index > Cubic Centimeter Definition

Cube root and square root is a topic of quantitative aptitude which questions are frequently asked in the competitive exams. If you are preparing for SSC, Banking or any other competitive exam, so here given cube root and square root formulas will be very useful for your preparation.

In this blog, you will get examples of cube root and square root and their solutions also as well as formulas. These examples, solutions, and formulas will help you to understand this topic easily.

Practice more with Square Root and Cube Root Problems with Solutions to obtain good marks in the competitive exams.

Formula and Methods of Square and Cube Root

If a number is multiple two with itself, then the result of this multiplication is called the cube of that number. E.g., cube of 6 = 6 × 6 × 6 = 216.

Definition of Cube Root

Step I Express the given number as the product of prime factors.

Step II Keep these factors in a group of three.

Step III Take the product of these prime factors picking one out of every group ( group of three) of the same primes. This product gives us the cube root of given number.

Ex. Find the cube root of 9261.

Prime factors of 9261

Now, taking one number from each group of three, we get

Important Trick

To find the cube of any number for two places after decimal

Let the given number be x, then take a number y in such a way that y should be near x and is a perfect cube.

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Definition of Square

If a number is multiplied with itself, then the result of this multiplication is called the square of that number. E.g, square of 6 = 6 × 6 = 36.

Definition of Square Root

Step II Keep these factors in pairs.

Step III Take the product of these prime factors taking one out of every pair of the same primes. This product gives us the square root of the given number.

Ex. Find the square root of 1089.

Prime number of 1089 = 11×11×3×3

Division Method of Square

Step I In given number, mark of the digit in pairs (also called period) starting from the unit’s digit.

Step II Divide the pair or period of the left side by a number whose square is equal to or less than that period. Write this number at the place of divisor and quotient and find the remainder.

Step III Bring down the next period and write to the left of remainder.

Step IV Write the double of quotient at the place of divisor. Now, write a digit to the right side of divisor which gives the products when multiplying with divisor, equal to or less than dividend.

Step V Write this digit is quotient also find the remainder.

Step VI Repeat the whole process till there is no period left over to be brought down. Thus, the quotient obtained will be the required square root of given number.

Important Tips for Division

1. Division method should be applied, when the given number is, so large that it is very difficult to find its square root by the prime factorisation method.

2. We can make a number which is not a perfect square, a perfect square by multiplying or dividing the number by those factors of it which are not contained in pairs.

3. A perfect square can never end with

● An odd number or zeroes.

● 2, 3, 7 and 8.

4. If in a number, number of digits is even (say k), then number of digits in its square root will be k/2 and if the number of digits is odd (say k), then the number of digits in its square root will be (k+1/2).

Example.  What is the square root of 151321?

⸫ Required square root = 389

To find the square root of any number for two places after decimal. Now, break the given number (say) x in two part (say) y and z such that one part (say) y is a perfect square, then

How to calculate the Square Root of Decimal Numbers ?

If in a given decimal number, the number of digital after decimal are not even, then we put a 0 at the extreme right.

Now we mark off the periods and try to calculate the square root applying the division method mentioned above.

Ex. Find the square root of 147.1369.

⸫ Required square root = 12.13

Examples with solutions

Q.1. Find the cube root of 2744

Resolve the given number as the product of prime factors and take the product of prime factors, choosing one out of three of the same prime factors. Resolving 2744 as the product of prime factors, we get:

2744 = 23×73

Q.2. By what least number 4320 be multiplied to obtain a number which is a perfect cube?

Solution .  Clearly, 4320 = 23×33×22×5

To make it a perfect cube, it must be multiplied by 2×52 = i.e. 50

Q.3. If the number p is 5 more than q and he sum of the squares of p and q is 55, then the product of p and q is

(B) – 10

(D) – 15

Solution .   According to the question

p-q = 5

On squaring both sides in Eq. (i), we get

(p-q)2 = 52

⸫ pq = 15.

Q.5. Each student of class X contributed some money for a picnic. The money contributed by each student was equal to the cube of the total number of students. If the total collected amount was Rs. 29791, then find the total number of students.

Solution .   Total collected amount = Rs. 29791.

⸫ Total number of students

Do you face difficulty to solve square root and cube root problems in the competitive exams or in your preparation? Here in this blog, I am sharing square root and cube root problems solutions with different examples.

You can learn easily how to solve your square root and cube root problems to score good marks with the help of these questions. First try to solve yourself square root and cube root problems before taking help examples.

Square Root and Cube Root Problems Solutions for Competitive Exams

(A)10.25               (B) 10.25               (C) 11.5                 (D) 19.5

Q.3. How many tow-digit numbers satisfy this property: The last digit (unit’s digit) of the square of the tow-digit number is 8 ?

(A) 1                       (B) 2                       (C) 3                       (D) None of these

A number ending in 8 can never be a perfect square.

Q.4. What is the square root of 0.16?

(A) 0.004                              (B) 0.04                 (C) 0.4                   (D) 4

(A) 0.00021                          (B) 0.0021                            (C) 0.021                               (D) 0.21

Q.6. A group of students decided to collect as many paise from each member of the group as is the number of members. If the total collection amounts to Rs. 59.29, the number of members in the group is:

(A) 57                    (B) 67                     (C) 77                     (D) 87

Q.7. The cube root of .000216 is :

(A) .6                     (B) .06                   (C) .006                (D) None of these

Q.9. The largest four-digit number which is perfect cube, is:

(A) 8000                (B) 9261                (C)9999                 (D) None of these

Clearly, 9261 is a perfect cube satisfying the given property.

Q.10. By what least number 675 be multiplied to obtain a number which is a perfect cube?

(A) 5                       (B) 6                       (C) 7                       (D) 8

675 = 5×5×3×3×3

To make it a a perfect cube, it must be multiplied by 5.

Q.11. What is the smallest number by which 3600 be divided to make it a perfect cube?

(A) 9                       (B) 50                     (C) 300                  (D) 450

To make it a perfect cube, it must be divided by 52×32×2 i.e, 450.

I hope these solutions are helpful for your competitive exams. If you have any query regarding square root and cube root problems solutions, you ask me in the comment section.

All the best!

Note: If no sign (or a positive sign) is placed in front of the square root, the positive answer is required. No sign means that a positive is understood. Only if a negative sign is in front of the square root is the negative answer required. This notation is used in many texts, as well as this book. Therefore,

To find the square root of a number that is not a perfect square, it is necessary to find an approximate answer by using the procedure given in Example .

Approximate  .

is between   and

6.5 × 6.5 = 42.25 or about 42.

Since   is slightly closer to   than it is to  ,

First, perform the operation under the radical.

Since   is slightly closer to   than it is to  .

Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind, because they are commonly used.

Sometimes you will have to simplify square roots, or write them in simplest form. In fractions,   can be simplified to  . In square roots,   can be simplified to  .

There are two main methods to simplify a square root.

Factor the number under the   into two factors, one of which is the largest possible perfect square. (Perfect squares are 1, 4, 9, 16, 25, 36, 49, and so on)

Completely factor the number under the   into prime factors and then simplify by bringing out any factors that came in pairs.

Simplify  .

Take the square root of the perfect square number

Finally, write it as a single expression.

Rewrite with pairs under the radical

In Example , the largest perfect square is easy to see, and Method 1 probably is a faster method.

In Example , it is not so obvious that the largest perfect square is 144, so method 2 is probably the faster method.

Remember: Most square roots cannot be simplified since they are already in simplest form, such as  ,  ,  .

Special note: If no sign (or a positive sign) is placed in front of the square root, then the positive answer is required. Only if a negative sign is in front of the square root is the negative answer required. This notation is used in many texts and is adhered to in this book. Therefore,

Approximating square roots

To find the square root of a number that is not a perfect square, it will be necessary to find an approximate answer by using the procedure given in Example

Approximate .

Since 62 = 36 and 72 = 49, then is between and .

Therefore, is a value between 6 and 7. Since 42 is about halfway between 36 and 49, you can expect that will be close to halfway between 6 and 7, or about 6.5. To check this estimation, 6.5 × 6.5 = 42.25, or about 42.

Square roots of nonperfect squares can be approximated, looked up in tables, or found by using a calculator. You may want to keep these two in mind:

Simplifying square roots

Sometimes you will have to simplify square roots, or write them in simplest form. In fractions, can be reduced to . In square roots, can be simplified to .

There are two main methods to simplify a square root.

Method 2: Completely factor the number under the into prime factors and then simplify by bringing out any factors that came in pairs.

Simplify .

Many square roots cannot be simplified because they are already in simplest form, such as , , and .

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