Rational exponents activity - Free Printable

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. It involves imaginary numbers, but since we are working with real numbers, this is undefined in the real number system.

Answer:
- Radical Form: \( \sqrt{-64} \)
- Simplified: Undefined (in the real number system)

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

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