# Cubing fractions

In this chapter we will learn to calculate the cube of fractions with solved examples.

To understand the chapter, you should have basic knowledge of fractions. Click the red link to review the fraction basics.

To revise the basics of cubes of numbers, click the red link.

## Representing cubing fraction

In cube of number, we multiply the number by itself three times.

If \mathtt{\frac{a}{b}} is the fraction, then the cube of fraction is represented as \mathtt{\left(\frac{a}{b}\right)^{3}}

## How to calculate cube of numbers ?

Let \mathtt{\frac{a}{b}} be the given fraction.

(a) Multiply the fraction by itself three times.

(b) Multiply the numerator and denominator separately

(c) If possible, simplify the resultant fraction.

The above process can be generalized as follows;

\mathtt{\left(\frac{a}{b}\right)^{3} \Longrightarrow \frac{a\times a\times a}{b\times b\times b} \Longrightarrow \frac{a^{3}}{b^{3}} \ }

I hope you understood the above process, let us now solve some problems.

Example 01
Find the cube of fraction 2 / 3

Solution
\mathtt{\Longrightarrow \left(\frac{2}{3}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 2\times 2}{3\times 3\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8}{27}}

Hence, 8/27 is the cube of given fraction.

Example 02
Find the cube of fraction 7 / 13

Solution
\mathtt{\Longrightarrow \left(\frac{7}{13}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 7\times 7}{13\times 13\times 13}}\\\ \\ \mathtt{\Longrightarrow \ \frac{343}{2197}}

Hence, 343 / 2197 is the cube of given fraction.

Example 03
Find the cube of fraction 3 / 5

Solution
Multiplying the same fractions thrice, we get;

\mathtt{\Longrightarrow \left(\frac{3}{5}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\times 3\times 3}{5\times 5\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{27}{125}}

Hence, 27 / 125 is the cube of given fraction.

Example 04
Find the cube of fraction \mathtt{\frac{-4}{9}}

Solution
If you want to understand the concept behind cubing of negative numbers, click the red link.

Cubing the above fraction;

\mathtt{\Longrightarrow \left(\frac{-4}{9}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-4\times -4\times -4}{9\times 9\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-64}{729}}

Hence, -64 / 729 is the cube of given fraction.

Example 05
Find the cube of fraction \mathtt{\frac{-11}{12}}

Solution
Multiply the same number three times;

\mathtt{\Longrightarrow \left(\frac{-11}{12}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-11\times -11\times -11}{12\times 12\times 12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-1331}{1728}}

Hence, -1331 / 1728 is the solution.

Example 06
Find the cube of fraction \mathtt{\frac{12}{15}}

Solution
The given fraction can be simplified further.

Divide the numerator and denominator by 3.

\mathtt{\Longrightarrow \ \frac{12\div 3}{15\div 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{12} \ 4}{\cancel{15} \ 5}}\\\ \\ \mathtt{\Longrightarrow \frac{4}{5}}

So the fraction has been reduced to 4 / 5

Now cubing the fraction, we get;

\mathtt{\Longrightarrow \left(\frac{4}{5}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4\times 4\times 4}{5\times 5\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{64}{125}}

Hence, 64 / 125 is the cube of fraction 12 / 15.

Example 07
Find the cube of fraction 7 / 343

Solution
The fraction can be simplified further.

Divide the numerator and denominator by 7

\mathtt{\Longrightarrow \ \frac{7\div 7}{343\div 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{7} \ 1}{\cancel{343} \ 49\ }}\\\ \\ \mathtt{\Longrightarrow \frac{1}{49}}

Now cubing the fraction 1/49, we get;

\mathtt{\Longrightarrow \left(\frac{1}{49}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1\times 1\times 1}{49\times 49\times 49}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{117649}}

Hence, 1 / 117649 is the cube of required fraction.

Example 08
Find the cube of fraction -4 / 14

Solution
The fraction can be simplified further.

Divide numerator and denominator by 2

\mathtt{\Longrightarrow \ \frac{-4\div 2}{14\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{-4} \ -2}{\cancel{14} \ 7\ }}\\\ \\ \mathtt{\Longrightarrow \frac{-2}{7}}

Now cubing the fraction -2 / 7, we get;

\mathtt{\Longrightarrow \left(\frac{-2}{7}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-2\times -2\times -2}{7\times 7\times 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-8}{343}}

Hence, -8 / 343 is the required solution.