# 8th Class Mathematics Cubes and Cube Roots Cube of a Real Number

## Cube of a Real Number

Category : 8th Class

### Introduction

The word cube is used in geometry. In geometry the word cube refers to the solid having equal sides. Thus cube of a natural number is the multiple of three prime factors of each number. A given number is said to be a perfect cube if it can be expressed as a product of triplets of equal factors.

### Cube of a Real Number

According to arithmetic and algebra, the cube of a number n is its third power. If a number multiplied three times by itself the resultant number is called cube of that number.

${{\text{n}}^{\text{3}}}=\text{n}\times \text{n}\times \text{n}$. In this expression if $\text{n}\times \text{n}\times \text{n}=\text{m}$ then we can say that m is cube of n. This is also the formula for volume of a geometric cube with sides of length "n".

The inverse operation of finding a number whose cube is 'n' is called finding the cube root of "n". It determines the side of the cube of a given volume.

Cubes of Certain Numbers which are Perfect Cube

${{1}^{3}}=1$                   ${{2}^{3}}=8$                  ${{3}^{3}}=27$                  ${{4}^{3}}=64$

${{5}^{3}}=125$              ${{6}^{3}}=216$              ${{7}^{3}}=343$               ${{8}^{3}}=512$

${{9}^{3}}=729$             ${{10}^{3}}=1000$          ${{11}^{3}}=1331$          ${{12}^{3}}=1728$

${{13}^{3}}=2197$         ${{14}^{3}}=2744$          ${{15}^{3}}=3375$          ${{16}^{3}}=4096$

${{17}^{3}}=4913$         ${{18}^{3}}=5832$           ${{19}^{3}}=6859$         ${{20}^{3}}=8000$

Cube of a Negative Number

We know that the cube of a negative number is always negative.

${{(-1)}^{3}}=-1\times -1\times -1=-1$

${{(-2)}^{3}}=-2\times -2\times -2=-8$

${{(-3)}^{3}}=-3\times -3\times -3=-27$

Cube Roots

The inverse operation of the cube of a number is called its cube root. It is normally denoted by. $\sqrt[3]{n}$ or ${{(n)}^{\frac{1}{3}}}$. The cube root of a number can be found by using the prime factorization method.

A number is called cube root of its cube.

The cube root of 8 is 2 because ${{2}^{3}}=2\times 2\times 2=8$

In symbolic form, the cube root of 8 is written as $\sqrt[3]{8}$

Likewise:

$\sqrt[3]{27}=3$            $(\because \,{{3}^{3}}=3\times 3\times 3=27)$

$\sqrt[3]{64}=4$            $(\because \,{{4}^{3}}=4\times 4\times 4=64)$

$\sqrt[3]{125}=5$          $(\because {{5}^{3}}=5\times 5\times 5=125)$

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