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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?).
Excel in math and science
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Evaluate \[\large \left(\frac{1}{256} \right)^{-\frac{5}{8 }}.\]
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by Brilliant Staff
If \( a>0 \), simplify
\[ \left( a^{\frac{2}{3}} + a^{-\frac{1}{3}} \right)^3 + \left( a^{\frac{2}{3}} - a^{-\frac{1}{3}} \right)^3. \]
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Evaluate
\[ \left( \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} \right)^{\frac13}. \]
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If \( x = \dfrac{3^{\frac{1}{5}} + 3^{ -\frac{1}{5}} }{2} \), evaluate
\[ \left( x + \sqrt{x^2 - 1} \right)^{10}. \]
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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 \(a+b\)?
Excel in math and science
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by Brilliant Staff