Question

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

Answer 1

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

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

2. We also note that \( 8 = 2^3 \), so:
\[
\sqrt{8} = (2^3)^{1/2} = 2^{3/2}
\]

3. Now we can rewrite the expression:
\[
\frac{3\sqrt{8}}{\sqrt{2}} = \frac{3 \cdot 2^{3/2}}{2^{1/2}}
\]

4. Apply the properties of exponents for division, specifically \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{3 \cdot 2^{3/2}}{2^{1/2}} = 3 \cdot 2^{3/2 - 1/2} = 3 \cdot 2^{(3/2 - 1/2)} = 3 \cdot 2^{2/2} = 3 \cdot 2^1 = 3 \cdot 2
\]

5. Finally, simplify:
\[
3 \cdot 2 = 6
\]

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

Answer 2

as a fraction