rewrite $sqrt5{4}cdotsqrt{2}$ as a single radical.\n$sqrt10{2^{3}}$\n$sqrt10{2^{9}}$\n$sqrt9{2^{10}}$\ndone

Explanation:

Step1: Rewrite radicals as exponents

$\sqrt[5]{4}\cdot\sqrt{2}=4^{\frac{1}{5}}\cdot2^{\frac{1}{2}}$. Since $4 = 2^2$, then $4^{\frac{1}{5}}=(2^2)^{\frac{1}{5}} = 2^{\frac{2}{5}}$. So the expression becomes $2^{\frac{2}{5}}\cdot2^{\frac{1}{2}}$.

Step2: Use exponent - product rule

According to the rule $a^m\cdot a^n=a^{m + n}$, we have $2^{\frac{2}{5}}\cdot2^{\frac{1}{2}}=2^{\frac{2}{5}+\frac{1}{2}}$. Calculate $\frac{2}{5}+\frac{1}{2}=\frac{4 + 5}{10}=\frac{9}{10}$. So the expression is $2^{\frac{9}{10}}$.

Step3: Rewrite exponent as radical

$2^{\frac{9}{10}}=\sqrt[10]{2^{9}}$.

Answer:

$\sqrt[10]{2^{9}}$