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 can break it down into several steps.

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

2. **Substitute \(\sqrt{256}\) into the expression**:
\[
(10 \sqrt{256})^5 = (10 \cdot 16)^5 = (160)^5
\]

3. **Now look at \(512^{1/3}\)**:
First, let's express \(512\) as a power of \(2\). We know that:
\[
512 = 2^9
\]
So,
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^{3} = 8
\]

4. **Putting it all together**:
\[
\frac{(160)^5}{8}
\]

5. **Calculate \(160^5\)**:
We can express \(160\) as \(16 \times 10\) or use direct multiplication;
\[
160^5 = (10 \cdot 16)^5 = 10^5 \cdot 16^5
\]

6. **Determine \(10^5\) and \(16^5\)**:
\[
10^5 = 100000
\]
For \(16^5\), we can calculate it step-by-step:
\[
16^2 = 256 \quad (16 \cdot 16)
\]
\[
16^4 = 256^2 = 65536 \quad (256 \cdot 256)
\]
\[
16^5 = 16^4 \cdot 16 = 65536 \cdot 16 = 1048576
\]

Now we find:
\[
160^5 = 10^5 \cdot 16^5 = 100000 \cdot 1048576
\]

7. **Finally simplify the expression**:
\[
\frac{160^5}{8}
\]

From \(100000 \cdot 1048576\), we calculate and divide by \(8\):
\[
8 = 2^3
\]

You can numerically calculate or simply keep it as is, but the final answer for \(\frac{160^5}{8}\) can retain either form as needed.

So the expression simplifies to:

\[
\frac{160^5}{8} \text{ or } 20 \cdot (16^5)
\]

In symbolic form, or you could use numerical approximations depending on your needs.