To simplify the expression [tex]\(\sqrt{\frac{1}{14}}\)[/tex], follow these steps:
1. Understand the structure of the expression: We have a square root of a fraction, specifically [tex]\(\sqrt{\frac{1}{14}}\)[/tex].
2. Rewrite the square root of a fraction: Recall that the square root of a fraction can be expressed as the square root of the numerator divided by the square root of the denominator:
[tex]\[
\sqrt{\frac{1}{14}} = \frac{\sqrt{1}}{\sqrt{14}}
\][/tex]
3. Simplify the square roots in the numerator and the denominator:
[tex]\(\sqrt{1}\)[/tex] is 1, because 1 is a perfect square and [tex]\(\sqrt{1} = 1\)[/tex].
[tex]\(\sqrt{14}\)[/tex] is an irrational number and remains as [tex]\(\sqrt{14}\)[/tex].
Therefore, the expression becomes:
[tex]\[
\frac{1}{\sqrt{14}}
\][/tex]
4. Rationalize the denominator (optional but recommended): To rationalize the denominator, multiply the numerator and denominator by [tex]\(\sqrt{14}\)[/tex]:
[tex]\[
\frac{1}{\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = \frac{\sqrt{14}}{14}
\][/tex]
5. Final simplified expression: Thus, the simplified form of the given expression is:
[tex]\[
\frac{\sqrt{14}}{14}
\][/tex]
6. Calculate the numerical value: Evaluating the numerical value of [tex]\(\frac{\sqrt{14}}{14}\)[/tex], we get approximately:
[tex]\[
0.2672612419124244
\][/tex]
So, the simplified expression [tex]\(\sqrt{\frac{1}{14}}\)[/tex] results in [tex]\(\frac{\sqrt{14}}{14}\)[/tex], which numerically approximates to [tex]\(0.2672612419124244\)[/tex].
