Cube Root of 2

The process of cubing is similar to squaring, the only difference is that the number is multiplied three times in the case of cubing and in squaring the number is multiplied two times. The exponent used for cubes is 3, which is also denoted by the superscript ³. For example, cube of a number can be expressed as 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512.

We can say that the cube root is the inverse operation of cubing a number. The cube root symbol is ∛.

In Mathematics, the cube root of a number (a) is the number (b) which satisfies the equation written below.

a³  = b

It can also be written in the form of

a = ∛b.

In this article, we will discuss about the meaning of cube root, its properties, the value of cube root of 2, and how to find the cube root of 2. We have also provided some solved examples so that you have a better understanding of the concept.

What is the Meaning of Cube Root?

The cube root of a number ‘a’ is a number that is obtained when we multiply three times to get the original number (i.e., the number ‘a’ itself). 

Let’s consider the following examples:

If the cube of a number 6, or 6³, is equal to 216, then the cube root of 216, or ∛216, is equal to 6.

If the cube of a number 5, or 5³, is equal to 125, then the cube root of 125, or ∛125, is equal to 5.

If the cube of a number 4, or 4³, is equal to 64, then the cube root of 64, or ∛64, is equal to 4.

In case if we need to find the cube root of any large numbers, then any one of the following  methods can be used.

1. Prime factorization Method 

2. Long Division Method

3. Using Logarithms

Properties of Cube Root

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

For example, ∛125 = 5, ∛27 = 3.

• The cube root of all the even natural numbers is even. For example: ∛8 = 2, ∛64 = 4.

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

What is the Value of Cube Root of 2?

The value of the cube root of 2 is 1.259921 (rounded to 6 decimal places). It is expressed in the form of ∛2. The number 2 is not a perfect cube. Therefore, the general methods cannot be used to find the cube root of 2.

Value of Cube Root of (∛2) = 1.259921

How to Find the 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 2. Therefore, we are going to use the logarithm method to find the cube root of 2..

Using Logarithms

Take log on both the sides,

x = 2^1/3

So,  log(x) = 1/3∗log(2)

log(x) = ⅓∗log(2)

log(x) = ⅓∗0.30102999=0.100343

log(x) = ⅓∗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

Cube Root for Non-perfect Cubes

Finding cube root for non-perfect cubes is not easy. So, we have provided a list of cube roots from 1 to 20 which is composed of only non-perfect cubes.

Solved Examples

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, so we get 2 x 2 x 2 x 7 x 7 x 7

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

= 2x 7

= 14

Therefore,  \[\sqrt[3]{2744}\] = 14.

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

Solution:

Now,

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

= 2×2×3

= 12

Example 3: Find the value of ∛2 + ∛(−2).

Solution:

∛−2 = −∛2, which means that the cube root of −2 is equal to the negative of the cube root of 2.

Therefore, ∛2 + ∛(−2) = ∛2 − ∛2 = 0.