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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.