Use the properties of exponents and radicals to generate an equivalent expression

for $\frac {(\sqrt [10]{256})^{5}}{512^{\frac {1}{3}}}$ (1 point)

1 answer

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: \[ \boxed{2} \]