Exercises

1. Evaluate.
   1. \(81^\frac{3}{4}\)
   2. \(49^ \frac{3}{2}\)
   3. \(32^\frac{8}{5}\)
   4. \(1000^\frac{5}{3}\)
2. Evaluate.
   1. \(\left(\dfrac{1}{8} \right)^\frac{-4}{3}\)
   2. \(\left(\dfrac{1}{4} \right)^\frac{-7}{2}\)
   3. \(\left(\dfrac{16}{81} \right)^\frac{-3}{4}\)
   4. \(125^\frac{-2}{3}\)
3. Write each expression as a power with a positive exponent and then evaluate.
   1. \(\sqrt[3]{8^2}\)
   2. \(\sqrt[5]{32^6}\)
   3. \(\sqrt[4]{16^{-5}}\)
   4. \(\sqrt{\left(\dfrac{16}{25}\right)^3}\)
   5. \(\sqrt[3]{\left(\dfrac{-189}{7}\right)^{-2}}\)
4. Simplify. Write your answer with positive exponents.
   1. \(a^\frac{1}{2}a^\frac{-2}{3}\)
   2. \(\left(\dfrac{1}{a}\right)^\frac{1}{2}a^\frac{-2}{3}\)
   3. \(\left(\dfrac{1}{a} \right)^\frac{1}{3}\div a^\frac{-1}{3}\)
   4. \(\dfrac{\left(\frac{1}{a} \right)^\frac{2}{3}a^\frac{-1}{3}}{(-a)^\frac{1}{3}\div\left(\frac{-1}{a} \right)^\frac{-1}{3}}\)
5. Express each power with an exponent of \(12\) (i.e., \(a^{12}\)).
   1. \(5^3\)
   2. \(6^{24}\)
   3. \(11^{15}\)
   4. \(d^\frac{2}{3}\)
6. Simplify \(\left(\sqrt{\dfrac{8^{n+2}}{4^{2n-3}}}\right)^\frac{-4}{13}\) with a positive exponent and then evaluate when \(n=-1\).
7. The surface area of a sphere can be calculated using the formula 

   \(A=\pi^\frac{1}{3}(6V)^\frac{2}{3}\)

   where

   A typical basketball has a volume of approximately \(59~808 \) cm³. Calculate the surface area of a basketball rounded to the nearest cm².

   • \(A\) represents the surface area, and
   • \(V\) represents the volume.
8. Faith invests \($5000\) for \(20\) months. The interest rate on her investment is \(4\%\) per year. Calculate how much her investment will be worth after the \(20\) months given:

   \(F=P(1+i)^n\)

   where

   • \(F\) is the future value of the investment,
   • \(P\) is the present value of the investment,
   • \(i\) is the annual interest rate (as a decimal), and
   • \(n\) is the length of the investment in years.