the following work models how to write the quotient $\frac{sqrt3{4}}{sqrt{2}}$ using rational exponents…

the following work models how to write the quotient $\frac{sqrt3{4}}{sqrt{2}}$ using rational exponents. $\frac{sqrt3{4}}{sqrt{2}}=\frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{(2^{2})^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}$. note that, in order to use the quotient property of exponents, the bases must be the same. using the work shown on the left, what is $\frac{sqrt3{4}}{sqrt{2}}$ in simplest radical form? 2 $sqrt{2^{6}}$ $sqrt6{2}$ done

the following work models how to write the quotient $\frac{sqrt3{4}}{sqrt{2}}$ using rational exponents. $\frac{sqrt3{4}}{sqrt{2}}=\frac{4^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{(2^{2})^{\frac{1}{3}}}{2^{\frac{1}{2}}}=\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}$. note that, in order to use the quotient property of exponents, the bases must be the same. using the work shown on the left, what is $\frac{sqrt3{4}}{sqrt{2}}$ in simplest radical form? 2 $sqrt{2^{6}}$ $sqrt6{2}$ done

Answer

Explanation:

Step1: Rewrite radicals as exponents

We know that $\sqrt[3]{4}=4^{\frac{1}{3}}=(2^{2})^{\frac{1}{3}} = 2^{\frac{2}{3}}$ and $\sqrt{2}=2^{\frac{1}{2}}$. So $\frac{\sqrt[3]{4}}{\sqrt{2}}=\frac{2^{\frac{2}{3}}}{2^{\frac{1}{2}}}$.

Step2: Use quotient - property of exponents

According to the quotient - property of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$, where $a = 2$, $m=\frac{2}{3}$, and $n=\frac{1}{2}$. Then $2^{\frac{2}{3}-\frac{1}{2}}=2^{\frac{4 - 3}{6}}=2^{\frac{1}{6}}$.

Step3: Convert back to radical form

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

Answer:

$\sqrt[6]{2}$