Question

Use the properties of exponents and radicals to generate an equivalent expression for (256√10)^5/512^ 1/3 .(1 point

Answer 1

To simplify the expression \((256\sqrt{10})^{5}/(512^{1/3})\), we can break it down step-by-step.

1. **Rewrite \(256\) and \(512\) as powers of \(2\)**:
- \(256 = 2^8\)
- \(512 = 2^9\)

2. **Rewrite \(\sqrt{10}\)**:
- \(\sqrt{10} = 10^{1/2}\)

3. **Rewriting the original expression**:
\[
(256 \sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = (2^8)^5 \cdot (10^{1/2})^5 = 2^{40} \cdot 10^{5/2}
\]

4. **Now rewrite \(512^{1/3}\)**:
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8
\]

5. **Now substitute into our expression**:
\[
\frac{(256 \sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{8} = \frac{2^{40} \cdot 10^{5/2}}{2^3} = 2^{40-3} \cdot 10^{5/2} = 2^{37} \cdot 10^{5/2}
\]

6. **Final expression**:
\[
2^{37} \cdot 10^{5/2}
\]

Thus, the equivalent expression for \((256\sqrt{10})^5/512^{1/3}\) is:

\[
2^{37} \cdot 10^{5/2}
\]