Rational Exponents Worksheet with Answer Key

What's on the worksheet

1. Evaluate each power, or say why it is not a real number:
   343^{1/3}0.0081^{1/4}(-36)^{1/2}-\left(\dfrac{8}{125}\right)^{1/3}32^{3/5}
2. Evaluate without a calculator:
   36^{-1/2}32^{-4/5}\left(\dfrac{9}{4}\right)^{-3/2}-1000^{-2/3}(-27)^{-2/3}
3. Simplify, writing each answer with positive exponents only:
   a^{3/4} \cdot a^{1/6}\left(b^{2/5}\right)^{15/4}\dfrac{c^{1/2}}{c^{5/6}}\left(8x^{3/4}y^{-6}\right)^{2/3}\dfrac{\left(9p^4\right)^{1/2}}{p^{1/2}}
4. Rewrite with rational exponents and write each answer as a single radical.
   \sqrt[3]{x^2} \cdot \sqrt[4]{x}\dfrac{\sqrt{x}}{\sqrt[3]{x}}\sqrt[3]{x\sqrt{x}}\dfrac{\sqrt[3]{4}}{\sqrt[6]{2}}
5. Solve each equation over the real numbers and check every answer:
   x^{1/4} = 3x^{2/5} = 4x^{-2/3} = \dfrac{1}{9}3x^{3/4} = 24x^{3/2} = -8
6. Word problem: the surface area of an ice cube from its volume.