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Evaluate: \[125\sqrt[3]{\alpha^6} - \sqrt[3]{125 \alpha^6}\]

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#### Solution

Property:

For any two integers *a* and *b*

\[\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}\]

From the above property, we have:

\[125\sqrt[3]{a^6} - \sqrt[3]{125 a^6}\]

\[ = 125\sqrt[3]{a^6} - \left( \sqrt[3]{125} \times \sqrt[6]{a^6} \right)\]

\[= 125 \times a^2 - \left( 5 \times a^2 \right)\]

∴ ( \[\sqrt[3]{a^6} = \sqrt[3]{\left\{ a \times a \times a \right\} \times \left\{ a \times a \times a \right\}} = a \times a = a^2 and \sqrt[3]{125} = \sqrt[3]{5 \times 5 \times 5} = 5\])

\[= 125 a^2 - 5 a^2 \]

\[ = 120 a^2\]

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