Question

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

Answer 1

To simplify the expression \((10\sqrt{256})^5 / 512^{1/3}\), we will start with simplifying each part of the expression.

1. **Calculate \( \sqrt{256} \)**:
\[
\sqrt{256} = 16
\]
Therefore, we have:
\[
10\sqrt{256} = 10 \times 16 = 160
\]

2. **Now, raise \(160\) to the power of \(5\)**:
\[
(10\sqrt{256})^5 = 160^5
\]

3. **Now, simplify \(512^{1/3}\)**:
First, we note that \(512\) can be expressed as \(2^9\) (because \(2^9 = 512\)). Therefore:
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8
\]

4. **Put it all together**:
Now we can rewrite the original expression:
\[
\frac{160^5}{512^{1/3}} = \frac{160^5}{8}
\]

5. **Evaluate \(160^5/8\)**:
We can express \(160\) in terms of its prime factors:
\[
160 = 2^5 \times 5^1
\]
Therefore:
\[
160^5 = (2^5 \times 5^1)^5 = 2^{25} \times 5^5
\]

6. **Now simplify \(\frac{160^5}{8}\)**:
Since \(8 = 2^3\):
\[
\frac{160^5}{8} = \frac{2^{25} \times 5^5}{2^3} = 2^{25 - 3} \times 5^5 = 2^{22} \times 5^5
\]

7. **Final expression**:
Thus, the equivalent expression for \((10\sqrt{256})^5 / 512^{1/3}\) is:
\[
2^{22} \times 5^5
\]

This can also be written in exponential form or simplified further, but this is a valid equivalent expression:
\[
\boxed{2^{22} \times 5^5}
\]