which expression is equivalent to $\frac{sqrt{2}}{sqrt3{2}}$?\n$\frac{1}{4}$\n$sqrt6{2}$\n$sqrt{2}$\n$\frac{s…

Explanation:

Step1: Rewrite the radicals as exponents

We know that $\sqrt{2}=2^{\frac{1}{2}}$ and $\sqrt[3]{2}=2^{\frac{1}{3}}$. So the expression $\frac{\sqrt{2}}{\sqrt[3]{2}}$ can be written as $\frac{2^{\frac{1}{2}}}{2^{\frac{1}{3}}}$.

Step2: Use the exponent - division rule

The rule for dividing two numbers with the same base $a$ is $\frac{a^m}{a^n}=a^{m - n}$. Here $a = 2$, $m=\frac{1}{2}$ and $n=\frac{1}{3}$. Then $2^{\frac{1}{2}-\frac{1}{3}}=2^{\frac{3 - 2}{6}}=2^{\frac{1}{6}}$.

Step3: Rewrite the exponent as a radical

Since $a^{\frac{1}{n}}=\sqrt[n]{a}$, when $a = 2$ and $n = 6$, $2^{\frac{1}{6}}=\sqrt[6]{2}$.

Answer:

$\sqrt[6]{2}$