Question

Find the cube root of the rational numbers 10648/12167

Answer 1

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Answer 2

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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 roots by factors

On factorising 10648 into prime factors, we get:

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

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

$\[1068 =\left\{2 \times 2 \times 2 \times \right\}\times \left \{11 \times 11 \times 11 \times\right\}$

Now, taking one factors from each triple, we get

$\sqrt[3]{12167}}$ = $\ 2 \times 11 =22$

Also factorising 12167 into prime factor, we get

12167 = ${\23 \times 23 \times 23}}$

Now getting one factor from the triple, we get:

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

Now,

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

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

= $\frac{22}{23}$

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