Algebra
Negative and Rational 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 \(a+b\)?