Show that:
(i) $ \frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\sqrt[3]{\frac{729}{1000}} $
(ii) $ \frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\sqrt[3]{\frac{-512}{343}} $

To find: 

We have to show that

(i) \( \frac{\sqrt[3]{729}}{\sqrt[3]{1000}}=\sqrt[3]{\frac{729}{1000}} \)

(ii) \( \frac{\sqrt[3]{-512}}{\sqrt[3]{343}}=\sqrt[3]{\frac{-512}{343}} \)

Solution:

(i) LHS $=\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}$

$=\frac{\sqrt[3]{9 \times 9 \times 9}}{\sqrt[3]{10 \times 10 \times 10}}$

$=\frac{\sqrt[3]{9^{3}}}{\sqrt[3]{10^{3}}}$

$=\frac{9}{10}$

$=0.9$

RHS $=\sqrt[3]{\frac{729}{1000}}$

$=\sqrt[3]{\frac{9 \times 9 \times 9}{10 \times 10 \times 10}}$

$=\sqrt[3]{\frac{9}{10} \times \frac{9}{10} \times \frac{9}{10}}$

$=\sqrt[3]{(\frac{9}{10})^{3}}$

$=\frac{9}{10}$

$=0.9$

LHS $=$ RHS

Hence proved.

(ii) LHS $=\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}$

$=\frac{-\sqrt[3]{512}}{\sqrt[3]{343}}$

$=\frac{-\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}}{\sqrt[3]{7 \times 7 \times 7}}$

$=\frac{-\sqrt[3]{2^{3} \times 2^{3} \times 2^{3}}}{\sqrt[3]{7^{3}}}$

$=\frac{-2 \times 2 \times 2}{7}$

$=\frac{-8}{7}$

RHS $=\sqrt[3]{\frac{-512}{343}}$

$=-\sqrt[3]{\frac{512}{343}}$

$=-\sqrt[3]{\frac{8 \times 8 \times 8}{7 \times 7 \times 7}}$

$=-\sqrt[3]{(\frac{8}{7} \times \frac{8}{7} \times \frac{8}{7})}$

LHS $=$ RHS

Hence proved.

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Updated on: 10-Oct-2022

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