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 properties of exponents.

First, let's rewrite the numbers in terms of powers of 2:

1. \( 8 = 2^3 \)
2. \( 32 = 2^5 \)

Now, we can rewrite the square root:

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

Now, substituting these values back into the expression:

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

Next, we can add the exponents since the bases are the same:

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

To add \( 3 \) and \( 5/2 \), we convert \( 3 \) into a fraction with a denominator of \( 2 \):

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

Now, we add:

\[
\frac{6}{2} + \frac{5}{2} = \frac{11}{2}
\]

Thus, we have:

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

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

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