Question

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

Answer 1

To simplify the expression \((256\sqrt{10})^5 / 512^{1/3}\), we can follow these steps using properties of exponents and radicals.

### Step 1: First simplify each part of the expression.

1. **Calculate \(512^{1/3}\)**:
\[
512 = 2^9 \quad (\text{since } 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2)
\]
Therefore,
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8
\]

2. **Calculate \((256\sqrt{10})^5\)**:
We recognize that \(256 = 2^8\);
thus,
\[
256\sqrt{10} = 2^8 \cdot 10^{1/2}
\]
Now raising this to the 5th power,
\[
(256\sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = (2^8)^5 \cdot (10^{1/2})^5
\]
Calculating these,
\[
(2^8)^5 = 2^{40}, \quad (10^{1/2})^5 = 10^{5/2}
\]
Therefore,
\[
(256\sqrt{10})^5 = 2^{40} \cdot 10^{5/2}
\]

### Step 2: Now put these together in the original expression.
So far we have:
\[
\frac{(256\sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{8} = \frac{2^{40} \cdot 10^{5/2}}{2^3}
\]

### Step 3: Simplify the fraction:
Using the properties of exponents,
\[
\frac{2^{40}}{2^3} = 2^{40 - 3} = 2^{37}
\]
Thus, we simplify the expression to:
\[
2^{37} \cdot 10^{5/2}
\]

### Final Expression:
Now, we have:
\[
(256\sqrt{10})^5 / 512^{1/3} = 2^{37} \cdot 10^{5/2}
\]

So the final equivalent expression in simplest form is:
\[
\boxed{2^{37} \cdot 10^{5/2}}
\]