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

         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:

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

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

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