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## Exponential Functions

From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).

# 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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