Question

Use the properties of exponents and radicals to generate an equivalent expression for (√256)^5/512^1/3 (Make the 256 to the tenth root please)

Answer 1

To simplify the expression \((\sqrt{256})^5 / 512^{1/3}\) and express \(256\) as a tenth root, we can follow these steps:

1. **Simplify \(\sqrt{256}\)**:
\[
\sqrt{256} = 256^{1/2}
\]
Since \(256 = 2^8\),
\[
\sqrt{256} = (2^8)^{1/2} = 2^{8 \cdot \frac{1}{2}} = 2^4
\]

2. **Substituting back into the expression**:
\[
(\sqrt{256})^5 = (2^4)^5 = 2^{4 \cdot 5} = 2^{20}
\]

3. **Now simplify \(512^{1/3}\)**:
Since \(512 = 2^9\),
\[
512^{1/3} = (2^9)^{1/3} = 2^{9 \cdot \frac{1}{3}} = 2^3
\]

4. **Putting it all back together**:
Now we have our expression:
\[
\frac{(\sqrt{256})^5}{512^{1/3}} = \frac{2^{20}}{2^3}
\]

5. **Using the property of exponents**:
\[
\frac{2^{20}}{2^3} = 2^{20 - 3} = 2^{17}
\]

6. **Express \(256\) as a tenth root (reducing it to the equivalent expression)**:
We know \(256 = 2^8\), so \(256\) can also be expressed in terms of a tenth root:
\[
256 = (256^{1/10})^{10} = (2^8)^{1/10} = 2^{\frac{8}{10}} = 2^{0.8}
\]

Thus, an equivalent expression for \((\sqrt{256})^5 / 512^{1/3}\) can be rewritten as:
\[
\frac{(256^{1/10})^5}{512^{1/3}} = \frac{(2^{0.8})^5}{2^3}
\]

Calculating this:
\[
(2^{0.8})^5 = 2^{0.8 \times 5} = 2^{4}
\]
Now simplifying:
\[
\frac{2^{4}}{2^{3}} = 2^{4 - 3} = 2^{1} = 2
\]

Finally, the equivalent expression for \((\sqrt{256})^5 / 512^{1/3}\), using the properties and adjustments made, yields:
\[
2
\]