Question

Use the properties of exponents and radicals to generate an equivalent expression
for $\frac {(\sqrt [10]{256})^{5}}{512^{\frac {1}{3}}}$ (1 point)

Answer 1

To simplify the expression \(\frac {(\sqrt [10]{256})^{5}}{512^{\frac {1}{3}}}\), we start by simplifying the components in the numerator and denominator.

1. **Calculating \(\sqrt[10]{256}\) and \((\sqrt[10]{256})^{5}\)**:
\[
256 = 2^8
\]
Therefore,
\[
\sqrt[10]{256} = 256^{\frac{1}{10}} = (2^8)^{\frac{1}{10}} = 2^{\frac{8}{10}} = 2^{\frac{4}{5}}
\]
Next, we raise this result to the power of 5:
\[
(\sqrt[10]{256})^5 = (2^{\frac{4}{5}})^5 = 2^{\frac{4}{5} \cdot 5} = 2^{4} = 16
\]

2. **Calculating \(512^{\frac{1}{3}}\)**:
\[
512 = 2^9
\]
Thus,
\[
512^{\frac{1}{3}} = (2^9)^{\frac{1}{3}} = 2^{9 \cdot \frac{1}{3}} = 2^{3} = 8
\]

3. **Combining the results**:
Now we can substitute back into our original expression:
\[
\frac{(\sqrt[10]{256})^5}{512^{\frac{1}{3}}} = \frac{16}{8}
\]

4. **Simplifying the fraction**:
\[
\frac{16}{8} = 2
\]

Thus, the equivalent expression for \(\frac {(\sqrt [10]{256})^{5}}{512^{\frac {1}{3}}}\) is:
\[
\boxed{2}
\]