Certainly! Let's solve the expression step-by-step.
Given the expression:
[tex]\[ \sqrt{\frac{1}{a} + \frac{1}{a}} \][/tex]
1. Simplify the expression inside the square root:
We start by combining the terms inside the square root.
[tex]\[ \frac{1}{a} + \frac{1}{a} \][/tex]
Since these are like terms, we can add them together:
[tex]\[ \frac{1}{a} + \frac{1}{a} = \frac{1}{a} \cdot 2 = \frac{2}{a} \][/tex]
2. Take the square root of the simplified expression:
Now our expression becomes the square root of [tex]\(\frac{2}{a}\)[/tex]:
[tex]\[ \sqrt{\frac{2}{a}} \][/tex]
3. Simplify the square root:
We can rewrite the square root of a quotient as the quotient of the square roots:
[tex]\[ \sqrt{\frac{2}{a}} = \frac{\sqrt{2}}{\sqrt{a}} \][/tex]
4. Combine the radicals:
To make it look neater, we can combine the radicals if needed, but in this case, [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{a}\)[/tex] are already in their simplest forms.
Finally, our expression simplifies to:
[tex]\[ \sqrt{2} \cdot \frac{1}{\sqrt{a}} = \frac{\sqrt{2}}{\sqrt{a}} \][/tex]
We can alternatively write this as:
[tex]\[ \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} \][/tex]
Thus, the result of the given expression is:
[tex]\[ \sqrt{2} \cdot \sqrt{\frac{1}{a}} \][/tex]
Or more compactly,
[tex]\[ \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} \][/tex]
This is our final answer:
[tex]\[ \sqrt{2 \cdot \frac{1}{a}} = \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \sqrt{1/a} \][/tex]
So, the simplified expression is:
[tex]\[ \sqrt{2} \cdot \sqrt{\frac{1}{a}} = \sqrt{2} \cdot \sqrt{1/a}= \sqrt{2} \cdot \sqrt{1/a}= \sqrt{2} \cdot \sqrt{1/a} = \sqrt{2} \cdot \sqrt{1/a}. \][/tex]
This is the final answer.
