Certainly! Let's go through it step-by-step to express [tex]\( 3^{-\frac{1}{2}} x^{\frac{1}{2}} \)[/tex] with radical signs and then rationalize the denominator.
### Step 1: Express Each Part with Radical Signs
First, convert the exponents into radical (square root) form:
1. For [tex]\( x^{\frac{1}{2}} \)[/tex]:
[tex]\[
x^{\frac{1}{2}} = \sqrt{x}
\][/tex]
2. For [tex]\( 3^{-\frac{1}{2}} \)[/tex]:
- The negative exponent indicates the reciprocal.
- [tex]\( 3^{-\frac{1}{2}} \)[/tex] can be written as [tex]\( \frac{1}{3^{\frac{1}{2}}} \)[/tex], which is [tex]\( \frac{1}{\sqrt{3}} \)[/tex].
Now, combining these, we have:
[tex]\[
3^{-\frac{1}{2}} x^{\frac{1}{2}} = \frac{\sqrt{x}}{\sqrt{3}}
\][/tex]
### Step 2: Rationalize the Denominator
To rationalize the denominator, we aim to eliminate the square root from the denominator. We achieve this by multiplying the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
1. Multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{\sqrt{x}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}
\][/tex]
2. Simplify the expression:
[tex]\[
\frac{\sqrt{x} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3x}}{3}
\][/tex]
### Final Answer
Thus, the expression [tex]\( 3^{-\frac{1}{2}} x^{\frac{1}{2}} \)[/tex] expressed with radical signs and with the denominator rationalized is:
[tex]\[
\frac{\sqrt{3x}}{3}
\][/tex]
