Question

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

Answer 1

To simplify the expression \((10\sqrt{256})^5 / 512^{1/3}\), we will work with both the numerator and the denominator separately.

1. **Simplify the numerator \((10\sqrt{256})^5\):**
- First, find \(\sqrt{256}\):
\[
\sqrt{256} = 16.
\]
- Now, substitute back into the expression:
\[
10\sqrt{256} = 10 \times 16 = 160.
\]
- Now raise \(160\) to the power of \(5\):
\[
(160)^5.
\]

2. **Simplify the denominator \(512^{1/3}\):**
- First, express \(512\) as a power of \(2\):
\[
512 = 2^9.
\]
- Now calculate \(2^9\) raised to the power of \(\frac{1}{3}\):
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8.
\]

3. **Combine the simplified forms:**
Now we have:
\[
\frac{(160)^5}{8}.
\]

4. **Further simplification can be performed**:
- Write \(160\) as \(2^5 \times 5\):
\[
160 = 2^5 \times 5.
\]
- Now raise this to the \(5\):
\[
(160)^5 = (2^5 \times 5)^5 = 2^{25} \times 5^5.
\]
- Therefore, the expression now looks like:
\[
\frac{2^{25} \times 5^5}{8}.
\]
- Replace \(8\) with \(2^3\):
\[
\frac{2^{25} \times 5^5}{2^3} = 2^{25-3} \times 5^5 = 2^{22} \times 5^5.
\]

5. **Final Equivalent Expression:**
The equivalent expression for \((10\sqrt{256})^5 / 512^{1/3}\) is:
\[
2^{22} \times 5^5.
\]