Cube Roots

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 \[\sqrt[3]{}\].

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 ‘32’. 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

\[\sqrt[3]{2}\] = x, then x3 =2 and we need to find here the value of x.

Value of Cube Root of

2(\[\sqrt[3]{2}\])   = 1.259921

What is the Meaning of Cube Root?

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

[Image- Cube Roots Example]

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 3  or \[\sqrt[3]{a}\]

Thus, the cube root of 125 is represented as \[\sqrt[3]{125}\] = 5 and that of 27 is represented as

\[\sqrt[3]{27}\]   = 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. 

For Example:

If the cube of a number 63 = 216

Then the cube root of 3√216 is equal to 6.

Cube root of any large numbers can be easily found in four ways:

Let’s know how to find the cube root of 2.

1. Prime factorization Method 

2. Long Division Method

3. Using Logarithms

4. Bisection Method

Properties of Cube Root

• The cube root of all the odd numbers is an odd number. 

For Example \[\sqrt[3]{125}\] = 5,

\[\sqrt[3]{27}\]   = 3.

• 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.

How to Find Cube Root of 2?

Let’s know how to find the cube root of 2. If n is a perfect cube for any integer namely m i.e., n = m³, then m can be known as the cube root of n and it can be denoted by m is equal to ∛n.

Since the number 2 is not a perfect cube, hence we cannot use here the prime factorization method or estimation method to find the cubic root of 4. Therefore, we are going to use another method here which is called the Newton Raphson method.

Follow the Steps Given below:

Step 1: Let us assume a number, suppose n, which is equal to the cube root of 2.

Here let us consider n=1.2 equal to the cube root of 2.

Step 2: Now, divide 2 by n and then divide its quotient again by n.

2/1.2 = 1.667 (first quotient)

and 1.667/2 = 1.389(second quotient)

Step 3: Take the average of n and the two quotients obtained by the division method. This value will be almost nearer to the value of

\[\sqrt[3]{2}\]  .

Hence, we get here three numbers to generate the average.

The number 1.2 (assumed number), 1.667 (first quotient), and 1.389 (second quotient).

(1.2+1.667+1.389)/3 

= 1.26

This value is almost near to the actual value of

\[\sqrt[3]{2}\]  , i.e. value of cube root of 2 (1.26)

Using Logarithms

Take log on both the sides,

x=2^1/3

So,  log(x)=1/3∗log(2)

log(x)=1/3∗log(2)

log(x)=1/3∗0.30102999=0.100343

log(x)=1/3∗0.30102999=0.100343 (approx)

Therefore, 

x=antilog(0.100343)=1.2599

x=antilog(0.100343)=1.2599 (approx)

Value of cube root of 2 =1.2599

Questions to be Solved

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 equals to 2 x 7 x 2 x 2 x 7 x 7

Step 2: Form groups of three similar factors we get 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:

Now,

\[\sqrt[3]{1728}\]=\[\sqrt[3] {2\times2\times2\times2\times2\times2\times3\times3\times3}\]

= 2 x 2 x 3

= 12

FAQs (Frequently Asked Questions)