**Category : **8th Class

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.

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:

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

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

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

*play_arrow*Cube of a Real Number*play_arrow*Cube Root of a Negative Number

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