Given:
[tex]3^{-\frac{1}{2}}.x^{\frac{1}{2}}[/tex]
To find:
Express with radical signs instead of fractional exponents. also, rationalize the denominator.
Explanation:
The radical sign is a symbol used to indicate a root, i.e.,
[tex]\sqrt[n]{x}[/tex]
For our given expression, we can write it using the radical sign as given below:
[tex]\begin{gathered} \frac{x^{\frac{1}{2}}}{3^{\frac{1}{2}}} \\ \Rightarrow\frac{\sqrt{x}}{\sqrt{3}} \end{gathered}[/tex]
Now, to rationalize, the following form can be used,
[tex]\frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{a}}{\sqrt{b}}(\frac{\sqrt{b}}{\sqrt{b}})=\frac{\sqrt{ab}}{b}[/tex]
So, we can also rewrite our expression to rationalize the denominator,
[tex]\frac{\sqrt{x}}{\sqrt{3}}=\frac{\sqrt{x}}{\sqrt{3}}\times(\frac{\sqrt{3}}{\sqrt{3}})=\frac{\sqrt{3x}}{3}[/tex]
Final answer:
The required expression with radical signs and simplified form is as given below:
[tex]\frac{\sqrt{3x}}{3}[/tex]
