Question

Fill in the blanks :

ix) $\sqrt[3]{3\frac{512}{\cdots }}=\frac{8}{13}$

Answer 1

Step: Fill in the blank

Given: $\sqrt[3]{3\frac{512}{\cdots }}=\frac{8}{13}$

Let the required number be x.

$\therefore \sqrt[3]{3\frac{512}{x}}=\frac{8}{13}$
$\Rightarrow \frac{\sqrt[3]{3512}}{\sqrt[3]{3x}}=\frac{8}{13}$

$(\because \sqrt[3]{3\frac{a}{b}}=\frac{\sqrt[3]{3a}}{\sqrt[3]{3b}})$


$\Rightarrow \frac{\sqrt[3]{32\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}}{\sqrt[3]{3x}}=\frac{8}{13}$
$\Rightarrow \frac{2\times 2\times 2}{\sqrt[3]{3x}}=\frac{8}{13}$
$\Rightarrow \sqrt[3]{3x}=13$
$\therefore x=(13{)}^3=2197$

Hence, the correct answer is $2197$.