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 first simplify each part step by step.

1. **Simplify \(\sqrt{256}\)**:
\[
\sqrt{256} = 16
\]

So, we can rewrite the expression:
\[
(10\sqrt{256})^5 = (10 \cdot 16)^5 = (160)^5
\]

2. **Calculate \((160)^5\)**:
At this stage, we can just keep it as \((160)^5\) for now.

3. **Simplify \(512^{1/3}\)**:
We can rewrite \(512\) as a power of \(2\):
\[
512 = 2^9
\]
Thus,
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8.
\]

4. **Put it all together**:
Now we substitute everything back into the original expression:
\[
\frac{(160)^5}{8}
\]

5. **Simplify the expression**:
We want to simplify \(\frac{(160)^5}{8}\):
\[
160 = 8 \cdot 20
\]
Thus,
\[
(160)^5 = (8 \cdot 20)^5 = 8^5 \cdot 20^5
\]

Now substituting this in gives us:
\[
\frac{8^5 \cdot 20^5}{8} = \frac{8^5}{8} \cdot 20^5 = 8^{5-1} \cdot 20^5 = 8^4 \cdot 20^5
\]

6. **Final Answer**:
We can write this final expression as:
\[
8^4 \cdot 20^5
\]

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

\[
8^4 \cdot 20^5
\]