Question

Find the cube root of the following rational number \[\frac{10648}{12167}\] .

Answer 1

Let us consider the following rational number: \[\frac{10648}{12167}\]

Now

\[\sqrt[3]{\frac{10648}{12167}}\]

\[= \frac{\sqrt[3]{10648}}{\sqrt[3]{12167}}\] ( ∵ \[\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\] ) 

Cube root by factors:
On factorising 10648 into prime factors, we get:

\[10648 = 2 \times 2 \times 2 \times 11 \times 11 \times 11\]

On grouping the factors in triples of equal factors, we get:

\[10648 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 11 \times 11 \times 11 \right\}\]

Now, taking one factor from each triple, we get:

\[\sqrt[3]{10648} = 2 \times 11 = 22\]

Also

On factorising 12167 into prime factors, we get:

\[12167 = 23 \times 23 \times 23\]

On grouping the factors in triples of equal factors, we get:

\[12167 = \left\{ 23 \times 23 \times 23 \right\}\]

Now, taking one factor from the triple, we get:

 

\[\sqrt[3]{12167} = 23\]

Now

\[\sqrt[3]{\frac{10648}{12167}}\]

\[= \frac{\sqrt[3]{10648}}{\sqrt[3]{12167}}\]

\[= \frac{22}{23}\]