Question

Use the properties of exponents to generate an equivalent expression for 3√8/√2.
2□

Answer 1

To simplify the expression \( \frac{3\sqrt{8}}{\sqrt{2}} \) using the properties of exponents, we can start by rewriting the square roots in terms of exponents:

1. Rewriting the square roots:
\[
\sqrt{8} = 8^{1/2}, \quad \text{and} \quad \sqrt{2} = 2^{1/2}
\]

2. Rewriting \( 8 \) in terms of powers of \( 2 \):
\[
8 = 2^3
\]
Therefore,
\[
\sqrt{8} = (2^3)^{1/2} = 2^{3/2}
\]

3. Substitute back into the original expression:
\[
\frac{3\sqrt{8}}{\sqrt{2}} = \frac{3 \cdot 2^{3/2}}{2^{1/2}}
\]

4. Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{2^{3/2}}{2^{1/2}} = 2^{3/2 - 1/2} = 2^{2/2} = 2^1 = 2
\]

5. Now, substitute this back:
\[
3 \cdot \frac{2^{3/2}}{2^{1/2}} = 3 \cdot 2 = 6
\]

Thus, the equivalent expression for \( \frac{3\sqrt{8}}{\sqrt{2}} \) is:
\[
6
\]

So the final expression is \( 6 \).