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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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