Square Root and Cube Root

Square Root and Cube Root Definition

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 side2, 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.


Similarly, when we 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 square root and cube root symbols are \[\sqrt{}\]and ∛ respectively.


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

Square Root

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, 22 =4, or the square root of 4 is 2

32 =9, or the square root of 9 is 3

42 = 16, or the square root of 16 is 4

The symbol of the square root is \[\sqrt{}\]

Therefore, the square root of 4 is represented as \[\sqrt{4}\] = 2.

And the square root of 9 is represented as \[\sqrt{9}\] = 3 and so on.


Cube Root

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

For Example, 23 =8, or the cube root of 8 is 2

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

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

The symbol of the cube root is a 1/3  or  ∛

Thus, the cube root of 8 is represented as \[\sqrt[3]{8}\] = 2 and that of 27 is represented as \[\sqrt[3]{27}\]= 3

and so on.


Properties of Square Root:

  • If the unit digit of a number is 2, 3, 7, and 8 then its square root is not a natural number.

  • If a number ends in an odd number of zeros, then its square root is not a natural number.

  • The square root of an even number is even and that of an odd number is odd.

  • Negative numbers have no squares root in a set of real numbers.


Properties of Cube Root

  • Cube root of all the odd numbers is an odd number. For example: \[\sqrt[3]{27}\]= 3, \[\sqrt[3]{125}\] = 5.

  • Cube root of all the even natural numbers is even. For example: \[\sqrt[3]{8}\] = 2,\[\sqrt[3]{64}\] = 4.

  • The cube root of a negative integer always results in negative.


Methods of Finding the Squares Roots and Cube Roots:

First, five number’s square root and cube root are easy to remember. But when we have to find square root and cube root of large numbers we can use the following methods to find it. They are

  1. Prime Factorisation Method

  2. Long Division method


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.


Square Root and Cube Root Table

Square Root of a Number

Number

Cube Root of a Number

Number

\[\sqrt{4}\]

2

\[\sqrt[3]{8}\]

2

\[\sqrt{9}\]

3

\[\sqrt[3]{27}\]

3

\[\sqrt{16}\]

4

\[\sqrt[3]{64}\]

4

\[\sqrt{25}\]

5

\[\sqrt[3]{135}\]

5

\[\sqrt{36}\]

6

\[\sqrt[3]{216}\]

6

\[\sqrt{49}\]

7

\[\sqrt[3]{343}\]

7

\[\sqrt{64}\]

8

\[\sqrt[3]{512}\]

8

\[\sqrt{81}\]

9

\[\sqrt[3]{729}\]

9

\[\sqrt{100}\]

10

\[\sqrt[3]{1000}\]

10

\[\sqrt{121}\]

11

\[\sqrt[3]{1331}\]

11

\[\sqrt{144}\]

12

\[\sqrt[3]{1728}\]

12

\[\sqrt{169}\]

13

\[\sqrt[3]{2197}\]

13

\[\sqrt{196}\]

14

\[\sqrt[3]{2744}\]

14

\[\sqrt{225}\]

15

\[\sqrt[3]{3375}\]

15

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


Solved Examples

Square root and Cube root example

Example 1: Find the cube root of 2744

Solution :

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.

= 23 x 73

= 143

Therefore, \[\sqrt[3]{2744}\]= 14

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

Solution:

2

1728

2

864

2

432

2

216

2

108

2

54

3

27

3

9

3

3


1

Now,

\[\sqrt[3]{1728}\]= \[\sqrt[3]{2X2X2X2X2X2X3X3}\]

= 2 x 2 x 3

= 12


Quiz

Some more square root and cube root examples to solve.

  1. Estimate the square root of 666

  2. Find the value of \[\sqrt{256}\]

  3. Using prime factorization, find the value of \[\sqrt[3]{1331}\]

  4. Using long division method, find the value of \[\sqrt[3]{729}\]

FAQ (Frequently Asked Questions)

What is the Difference Between the Square Root and Cube Root?

Answer:

  • The square root of a number x is that number which when multiplied by itself gives the number x itself.For example, the square root of 4 is 2.

  • The cube root of a number is a number which when multiplied thrice gives you an original number. For example, the cube root of 9 is 3.

  • The cube root of a number is the length of a cube whose volume is a3 units. Whereas the square root of a number is the length of a square whose area is a2 units.

  • The square root is an inverse operation of a squaring a number, while cube root is an inverse operation of cubing a number.

  • The square root is dented by √a, while cube root is denoted by ∛a.

2. What is the Formula for Cube Root?

Answer: We know that the cube of any number is found by multiplying that number three times. And the cube root of a number is the inverse operation of cubing a number. 

Example: If the cube of a number 53 = 125

Then cube root of ∛125 = 5

Cube root of largest number can be found in two  ways

Prime factorization method and Long Division method.