what is the following product?\n$sqrt5{4x^{2}}cdotsqrt5{4x^{2}}$\n$4x^{2}$\n$sqrt5{16x^{4}}$\n$2left(sqrt5{4x…

Explanation:

Step1: Use product - rule of radicals

For $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$ when $n$ is a positive integer. Here $n = 5$, $a = 4x^{2}$ and $b=4x^{2}$. $\sqrt[5]{4x^{2}}\cdot\sqrt[5]{4x^{2}}=\sqrt[5]{(4x^{2})\cdot(4x^{2})}$

Step2: Multiply the coefficients and variables

$(4x^{2})\cdot(4x^{2})=(4\times4)\cdot(x^{2}\cdot x^{2})$. Using the rule $a^{m}\cdot a^{n}=a^{m + n}$, we have $x^{2}\cdot x^{2}=x^{2 + 2}=x^{4}$ and $4\times4 = 16$. So $(4x^{2})\cdot(4x^{2})=16x^{4}$. Then $\sqrt[5]{(4x^{2})\cdot(4x^{2})}=\sqrt[5]{16x^{4}}$

Answer:

$\sqrt[5]{16x^{4}}$ (corresponding to the second option)