Algebra
# Radical Expressions and Equations

Evaluate

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

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 root on the right hand side.

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

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

Simplify

$\sqrt{20 + 2\sqrt{91}}.$