Worksheet for converting rational exponents to radical form and simplifying expressions.

To solve the problem, we need to convert each rational exponent into its radical form and then simplify the expression. Let's go through each row step by step.

1. \( 64^{\frac{1}{2}} \)

- Exponent Form: \( 64^{\frac{1}{2}} \)
- Radical Form: The exponent \(\frac{1}{2}\) means the square root. So, \( 64^{\frac{1}{2}} = \sqrt{64} \).
- Simplified: \( \sqrt{64} = 8 \).

Answer:
- Radical Form: \( \sqrt{64} \)
- Simplified: \( 8 \)

2. \( (-64)^{\frac{1}{3}} \)

- Exponent Form: \( (-64)^{\frac{1}{3}} \)
- Radical Form: The exponent \(\frac{1}{3}\) means the cube root. So, \( (-64)^{\frac{1}{3}} = \sqrt[3]{-64} \).
- Simplified: \( \sqrt[3]{-64} = -4 \) because \((-4)^3 = -64\).

Answer:
- Radical Form: \( \sqrt[3]{-64} \)
- Simplified: \( -4 \)

3. \( (-64)^{\frac{1}{2}} \)

- Exponent Form: \( (-64)^{\frac{1}{2}} \)
- Radical Form: The exponent \(\frac{1}{2}\) means the square root. So, \( (-64)^{\frac{1}{2}} = \sqrt{-64} \).
- Simplified: The square root of a negative number is not a real number. Therefore, it is undefined in the real number system.

Answer:
- Radical Form: \( \sqrt{-64} \)
- Simplified: Undefined

4. \( 32^{\frac{1}{5}} \)

- Exponent Form: \( 32^{\frac{1}{5}} \)
- Radical Form: The exponent \(\frac{1}{5}\) means the fifth root. So, \( 32^{\frac{1}{5}} = \sqrt[5]{32} \).
- Simplified: \( \sqrt[5]{32} = 2 \) because \(2^5 = 32\).

Answer:
- Radical Form: \( \sqrt[5]{32} \)
- Simplified: \( 2 \)

5. \( 16^{\frac{3}{4}} \)

- Exponent Form: \( 16^{\frac{3}{4}} \)
- Radical Form: The exponent \(\frac{3}{4}\) means the fourth root raised to the power of 3. So, \( 16^{\frac{3}{4}} = (\sqrt[4]{16})^3 \).
- Simplified: \( \sqrt[4]{16} = 2 \) because \(2^4 = 16\). Then, \(2^3 = 8\).

Answer:
- Radical Form: \( (\sqrt[4]{16})^3 \)
- Simplified: \( 8 \)

6. \( \left( \frac{100}{9} \right)^{-\frac{3}{2}} \)

- Exponent Form: \( \left( \frac{100}{9} \right)^{-\frac{3}{2}} \)
- Radical Form: The exponent \(-\frac{3}{2}\) means the reciprocal of the square root raised to the power of 3. So, \( \left( \frac{100}{9} \right)^{-\frac{3}{2}} = \left( \sqrt{\frac{9}{100}} \right)^3 \).
- Simplified: \( \sqrt{\frac{9}{100}} = \frac{3}{10} \). Then, \( \left( \frac{3}{10} \right)^3 = \frac{27}{1000} \).

Answer:
- Radical Form: \( \left( \sqrt{\frac{9}{100}} \right)^3 \)
- Simplified: \( \frac{27}{1000} \)

7. \( -27^{-\frac{4}{3}} \)

- Exponent Form: \( -27^{-\frac{4}{3}} \)
- Radical Form: The exponent \(-\frac{4}{3}\) means the reciprocal of the cube root raised to the power of 4. So, \( -27^{-\frac{4}{3}} = -\left( \sqrt[3]{27} \right)^{-4} \).
- Simplified: \( \sqrt[3]{27} = 3 \). Then, \( 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \). Therefore, \( -\left( \frac{1}{81} \right) = -\frac{1}{81} \).

Answer:
- Radical Form: \( -\left( \sqrt[3]{27} \right)^{-4} \)
- Simplified: \( -\frac{1}{81} \)

8. \( 49^{-\frac{3}{2}} \)

- Exponent Form: \( 49^{-\frac{3}{2}} \)
- Radical Form: The exponent \(-\frac{3}{2}\) means the reciprocal of the square root raised to the power of 3. So, \( 49^{-\frac{3}{2}} = \left( \sqrt{49} \right)^{-3} \).
- Simplified: \( \sqrt{49} = 7 \). Then, \( 7^{-3} = \frac{1}{7^3} = \frac{1}{343} \).

Answer:
- Radical Form: \( \left( \sqrt{49} \right)^{-3} \)
- Simplified: \( \frac{1}{343} \)

Final Answer

\[
\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Exponent Form} & \text{Radical Form} & \text{Simplified} \\
\hline
64^{\frac{1}{2}} & \sqrt{64} & 8 \\
\hline
(-64)^{\frac{1}{3}} & \sqrt[3]{-64} & -4 \\
\hline
(-64)^{\frac{1}{2}} & \sqrt{-64} & \text{Undefined} \\
\hline
32^{\frac{1}{5}} & \sqrt[5]{32} & 2 \\
\hline
16^{\frac{3}{4}} & (\sqrt[4]{16})^3 & 8 \\
\hline
\left( \frac{100}{9} \right)^{-\frac{3}{2}} & \left( \sqrt{\frac{9}{100}} \right)^3 & \frac{27}{1000} \\
\hline
-27^{-\frac{4}{3}} & -\left( \sqrt[3]{27} \right)^{-4} & -\frac{1}{81} \\
\hline
49^{-\frac{3}{2}} & \left( \sqrt{49} \right)^{-3} & \frac{1}{343} \\
\hline
\end{array}
}
\]