Question

$1. \: \sqrt[3]{32 \times ( - 250)}$
$2. \: \sqrt[3]{ \frac{648}{3993} }$
$3. \: \sqrt[3]{ \frac{256}{54 \times686} }$
Need Explanation.

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Answer 1

$1. \: \sqrt[3]{32 \times ( - 250)}$

$\tt{ = \sqrt[3]{ -(32 \times 250)} }$

$\tt{ = - \sqrt[3]{32 \times 250} }$

$\tt{ = - \sqrt[3]{32\times(2 \times 125)} }$

$\tt{ = - \sqrt[3]{(32 \times 2) \times 125} }$

$\tt{ = - \sqrt[3]{64\times125}}$

$\tt{ = - \sqrt[3]{4^{3} \times {5}^{3} } }$

$\tt{ = - \sqrt[3]{ {4}^{3}} \times \sqrt[3]{ {5}^{3}} }$

$\tt{ = - 4 \times 5}$

$\tt{ = - 20}.$

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$2. \: \sqrt[3]{ \frac{648}{3993} }$

$\tt{= \sqrt[3]{ \frac{3 \times 216}{3 \times 1331} }}$

$\tt{ = \sqrt[3]{ \frac{216}{1331} } }$

$\tt{ = \frac{ \sqrt[3]{216} }{ \sqrt[3]{1331} } }$

$\tt{ = \frac{ \sqrt[3]{6^{3} } }{ \sqrt[3]{ {11}^{3} } } }$

$\tt{ = \frac{6}{11}.}$

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$3. \: \sqrt[3]{ \frac{256}{54 \times686} }$

$\tt{ = \sqrt[3]{ \frac{4 \times 64}{2 \times 27 \times 2 \times 343} } }$

$\tt{ = \sqrt[3]{ \frac{64}{27 \times 343} } }$

$\tt{ = \frac{ \sqrt[3]{64} }{ \sqrt[3]{27} \times \sqrt[3]{343} } }$

$\tt{ = \frac{ \sqrt[3]{ {4}^{3} } }{ \sqrt[3]{ {3}^{3} \times \sqrt[3]{ {7}^{3} } } } }$

$\tt{ = \frac{4}{3 \times 7}}$

$\tt{ = \frac{4}{21}.}$

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