use the properties of exponents to generate an equivalent expression for (\root(3)(8))/(\sqrt(2))

To simplify the expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) using the properties of exponents, we start by rewriting each component in terms of base \(2\).

1. **Rewrite \(\sqrt[3]{8}\)**:
\[
8 = 2^3 \implies \sqrt[3]{8} = (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2.
\]

2. **Rewrite \(\sqrt{2}\)**:
\[
\sqrt{2} = 2^{\frac{1}{2}}.
\]

Now we substitute these rewritten expressions into the original expression:
\[
\frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2}{2^{\frac{1}{2}}}.
\]

3. **Use the properties of exponents**:
Recall the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\).
So we have:
\[
\frac{2}{2^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}}.
\]

Thus, the equivalent expression is:
\[
2^{\frac{1}{2}}.
\]

Finally, we can summarize our finding:
\[
\frac{\sqrt[3]{8}}{\sqrt{2}} = 2^{\frac{1}{2}}.
\]