Algebra

# Fractional Exponents

Evaluate $\large \left(\frac{1}{256} \right)^{-\frac{5}{8 }}.$

If $a>0$, simplify

$\left( a^{\frac{2}{3}} + a^{-\frac{1}{3}} \right)^3 + \left( a^{\frac{2}{3}} - a^{-\frac{1}{3}} \right)^3.$

Evaluate

$\left( \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} \right)^{\frac13}.$

If $x = \dfrac{3^{\frac{1}{5}} + 3^{ -\frac{1}{5}} }{2}$, evaluate

$\left( x + \sqrt{x^2 - 1} \right)^{10}.$

Suppose that $a$ and $b$ satisfy $12 \times 24^{\frac{1}{3}}+81^{\frac{1}{3}}+6 \times \left(3 \times 2^{15}\right)^{\frac{1}{3}}=a \sqrt[3]{b},$ where $b$ is a prime number. What is $a+b$?

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