Question

Use the properties of exponents to generate an equivalent expression for 8√32√.

Answer 1

To simplify the expression \( 8\sqrt{32} \), we can use the properties of exponents.

First, we can rewrite the components using exponents:
- \( 8 \) can be expressed as \( 2^3 \).
- \( 32 \) can be expressed as \( 2^5 \).

Now, substituting these expressions into the original expression, we have:

\[
8\sqrt{32} = 2^3 \sqrt{32} = 2^3 \sqrt{2^5}
\]

Next, we can simplify \( \sqrt{32} \):
\[
\sqrt{32} = \sqrt{2^5} = 2^{5/2}
\]

Now substituting this back into the expression:

\[
8\sqrt{32} = 2^3 \cdot 2^{5/2}
\]

Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we combine the exponents:

\[
2^3 \cdot 2^{5/2} = 2^{3 + 5/2}
\]

To add the exponents, we convert \( 3 \) to a fraction with a denominator of 2:

\[
3 = \frac{6}{2}
\]

Now add \( \frac{6}{2} + \frac{5}{2} = \frac{11}{2} \).

Thus, we have:

\[
8\sqrt{32} = 2^{\frac{11}{2}}
\]

In conclusion, the equivalent expression for \( 8\sqrt{32} \) is:

\[
\boxed{2^{\frac{11}{2}}}
\]