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Negative and Rational Exponents

Fractional Exponents


Evaluate (1256)58.\large \left(\frac{1}{256} \right)^{-\frac{5}{8 }}.

If a>0 a>0 , simplify

(a23+a13)3+(a23a13)3. \left( a^{\frac{2}{3}} + a^{-\frac{1}{3}} \right)^3 + \left( a^{\frac{2}{3}} - a^{-\frac{1}{3}} \right)^3.


(2+323)13. \left( \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}} \right)^{\frac13}.

If x=315+3152 x = \dfrac{3^{\frac{1}{5}} + 3^{ -\frac{1}{5}} }{2} , evaluate

(x+x21)10. \left( x + \sqrt{x^2 - 1} \right)^{10}.

Suppose that aa and bb satisfy 12×2413+8113+6×(3×215)13=ab3,12 \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 bb is a prime number. What is a+ba+b?


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