Use the properties of exponents to generate an equivalent expression for ^3square root 8 over square root 2

To simplify the expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\), we can use properties of exponents.

1. First, rewrite the roots in exponential form:
\[
\sqrt[3]{8} = 8^{1/3} \quad \text{and} \quad \sqrt{2} = 2^{1/2}
\]

2. Next, express \(8\) as a power of \(2\):
\[
8 = 2^3
\]
Thus,
\[
\sqrt[3]{8} = (2^3)^{1/3} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2
\]

3. Now substitute \(2\) back into the original expression:
\[
\frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{2^{1/2}}
\]

4. Using the properties of exponents:
\[
\frac{2}{2^{1/2}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}}
\]

5. Finally, rewrite \(2^{1/2}\) in radical form:
\[
2^{\frac{1}{2}} = \sqrt{2}
\]

Putting it all together, the equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) is:
\[
\sqrt{2}
\]