simplify: $\frac{sqrt4{32}}{sqrt4{256}}$

Explanation:

Step1: Prime - factorize the numbers

Express 32 and 256 as products of prime factors. $32 = 2^5$ and $256=2^8$. So, $\sqrt[4]{32}=\sqrt[4]{2^5}$ and $\sqrt[4]{256}=\sqrt[4]{2^8}$.

Step2: Use the radical rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$

$\sqrt[4]{2^5}=2^{\frac{5}{4}}$ and $\sqrt[4]{2^8}=2^{\frac{8}{4}} = 2^2$.

Step3: Calculate the fraction

$\frac{\sqrt[4]{32}}{\sqrt[4]{256}}=\frac{2^{\frac{5}{4}}}{2^2}$. According to the rule $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{2^{\frac{5}{4}}}{2^2}=2^{\frac{5}{4}-2}=2^{\frac{5 - 8}{4}}=2^{-\frac{3}{4}}$. Also, we can write it as $\frac{1}{2^{\frac{3}{4}}}=\frac{1}{\sqrt[4]{2^3}}=\frac{1}{\sqrt[4]{8}}$.

Answer:

$\frac{1}{\sqrt[4]{8}}$