Question

What is the radical form of each of the given expressions?

Answer 1

The main rule to apply here is:

(i) $\displaystyle{ a^ {\displaystyle{ (\frac{b}{c})} }= \displaystyle{ \sqrt[c]{a^b}$

(ii)If c=2, then we write the following $\displaystyle{ a^ {\displaystyle{ (\frac{b}{2})} }= \displaystyle{ \sqrt{a^b}$.


According to these rules:

$\displaystyle{ 5^ {\displaystyle{ (\frac{2}{3})} }= \displaystyle{ \sqrt[3]{5^2}$.

$\displaystyle{ 5^ {\displaystyle{ (\frac{1}{2})} }= \displaystyle{ \sqrt{5^1}=\displaystyle{ \sqrt{5}$.

$\displaystyle{ 3^ {\displaystyle{ (\frac{2}{5})} }= \displaystyle{ \sqrt[5]{3^2}$.

$\displaystyle{ 3^ {\displaystyle{ (\frac{5}{2})} }= \displaystyle{ \sqrt{3^5}$.

Answer 2

Answer:$5^{\frac{2}{3}}= \sqrt[3]{5^2}$

$5^{\frac{1}{2}}= \sqrt{5}$

$3^{\frac{2}{5}}= \sqrt[5]{3^2}$

$3^{\frac{5}{2}}= \sqrt{3^5}$

Step-by-step explanation: According to the rule of exponent of radical form:

$a^{\frac{m}{n} } = \sqrt[n]{a^m}$.

Let us apply same rule in first number

$5^{\frac{2}{3}}= \sqrt[3]{5^2}$

$5^{\frac{1}{2}}= \sqrt{5}$

$3^{\frac{2}{5}}= \sqrt[5]{3^2}$

$3^{\frac{5}{2}}= \sqrt{3^5}$

Note: When we don't have any number on the top of radical, there 2 is understood.