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Combine your skills of working with exponents and manipulating expressions for radical glory.

Evaluate

\[ \sqrt[3]{8} \times \sqrt[4]{64} \div \sqrt{2}. \]

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What is the value of \(b\) that satisfies \[\sqrt{\frac{a^{\frac{1}{5}}}{a^{\frac{1}{7}}}} \times \sqrt[7]{\frac{a}{a^{\frac{1}{5}}}} \times \sqrt[5]{\frac{a^{\frac{1}{7}}}{a^{\frac{1}{2}}}}=\sqrt[b]{a}?\]

**Details and assumptions**

\(b\) is the index of the square root on the right hand side.

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What is the value of \( \left(13 + \sqrt{-169}\right)\left(10 - \sqrt{-100}\right) \)?

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If \[x=\sqrt[3]{10+\sqrt{127}}+\sqrt[3]{10-\sqrt{127}},\] what is the value of \(x^3+9x ?\)

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Simplify

\[ \sqrt{20 + 2\sqrt{91}}. \]

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