Question

convert from radical form to exponential expression in rational form no need to evaluate just be put in simplest form( i already know how to simplify the multiplication oart i j need help in the division)

Answer 1

In order to convert from radical to exponential form, we can use the property below:

$\sqrt[c]{a^b}=a^{\frac{b}{c}}$

Then, to find the simplest form, we use this property:

$\begin{gathered} a^b\cdot a^c=a^{b+c} \\ a^b\colon a^c=a^{b-c} \end{gathered}$

So we have:

$\begin{gathered} \sqrt[5]{x^3}=x^{\frac{3}{5}} \\ \sqrt[]{x^4}=x^{\frac{4}{2}} \\ \sqrt[]{x^3}=x^{\frac{3}{2}} \\ \\ \sqrt[5]{x^3}\cdot\sqrt[]{x^4}\colon\sqrt[]{x^3} \\ =x^{\frac{3}{5}}\cdot x^{\frac{4}{2}}\colon x^{\frac{3}{2}} \\ =x^{\frac{3}{5}+\frac{4}{2}-\frac{3}{2}} \\ =x^{\frac{3}{5}+\frac{1}{2}} \\ =x^{\frac{6}{10}+\frac{5}{10}} \\ =x^{\frac{11}{10}} \\ =x^1\cdot x^{\frac{1}{10}} \\ =x\sqrt[10]{x} \end{gathered}$