what is the simplified form of the following expression? assume x > 0. (sqrt4{\frac{3}{2x}})…

what is the simplified form of the following expression? assume x > 0. (sqrt4{\frac{3}{2x}}) (\frac{sqrt4{6x}}{2x}) (\frac{sqrt4{24x^{3}}}{2x}) (\frac{sqrt4{24x^{3}}}{16x^{4}}) (sqrt4{12x^{2}})

what is the simplified form of the following expression? assume x > 0. (sqrt4{\frac{3}{2x}}) (\frac{sqrt4{6x}}{2x}) (\frac{sqrt4{24x^{3}}}{2x}) (\frac{sqrt4{24x^{3}}}{16x^{4}}) (sqrt4{12x^{2}})

Answer

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of $\sqrt[4]{\frac{3}{2x}}$, we multiply the numerator and denominator inside the fourth - root by $2^{3}x^{3}$ (because we want the denominator to be a perfect fourth - power). So we have $\sqrt[4]{\frac{3\times2^{3}x^{3}}{2x\times2^{3}x^{3}}}=\sqrt[4]{\frac{24x^{3}}{2^{4}x^{4}}}$.

Step2: Simplify the fourth - root

$\sqrt[4]{\frac{24x^{3}}{16x^{4}}}=\frac{\sqrt[4]{24x^{3}}}{\sqrt[4]{16x^{4}}}$. Since $\sqrt[4]{16x^{4}} = 2x$ for $x>0$, the simplified form is $\frac{\sqrt[4]{24x^{3}}}{2x}$.

Answer:

$\frac{\sqrt[4]{24x^{3}}}{2x}$