Question

Use the properties of exponents and radicals to generate an equivalent expression for (10√256)^5/512^1/3 (It is 256 to the tenth root, not the square root of 256* 10)

Answer 1

To simplify the expression \((10\sqrt[10]{256})^5 / 512^{1/3}\), we can break it down step by step using the properties of exponents and radicals.

### Step 1: Simplify \(10\sqrt[10]{256}\)

First, we rewrite \(256\) in exponential form:
\[
256 = 2^8
\]
Thus, the tenth root of \(256\) can be expressed as:
\[
\sqrt[10]{256} = \sqrt[10]{2^8} = 2^{8/10} = 2^{4/5}
\]

Now, substituting back into our expression, we have:
\[
10\sqrt[10]{256} = 10 \cdot 2^{4/5}
\]

### Step 2: Raise the expression to the power of 5

Now let's raise \(10\sqrt[10]{256}\) to the power of 5:
\[
(10 \cdot 2^{4/5})^5 = 10^5 \cdot (2^{4/5})^5 = 100000 \cdot 2^{4} = 100000 \cdot 16 = 1600000
\]

### Step 3: Simplify \(512^{1/3}\)

Next, let's simplify \(512^{1/3}\). We can express \(512\) as:
\[
512 = 2^9
\]
Thus:
\[
512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^{3} = 8
\]

### Step 4: Combine the two parts

Now we can substitute back and combine the two results:
\[
\frac{(10\sqrt[10]{256})^5}{512^{1/3}} = \frac{1600000}{8}
\]

Calculating this gives:
\[
\frac{1600000}{8} = 200000
\]

### Final Equivalent Expression

Thus, the equivalent expression for \((10\sqrt[10]{256})^5 / 512^{1/3}\) is:
\[
\boxed{200000}
\]