To express the fraction [tex]\(\frac{1}{20 \sqrt{20}}\)[/tex] in the form [tex]\(\frac{\sqrt{5}}{n}\)[/tex], where [tex]\( n \)[/tex] is a positive integer, we need to rationalize the denominator and simplify the fraction. Here's how you can do it step by step:
1. Starting with the given fraction:
[tex]\[
\frac{1}{20 \sqrt{20}}
\][/tex]
2. Rationalize the denominator:
To rationalize the denominator, we multiply both the numerator and the denominator by [tex]\(\sqrt{20}\)[/tex]:
[tex]\[
\frac{1}{20 \sqrt{20}} \times \frac{\sqrt{20}}{\sqrt{20}} = \frac{\sqrt{20}}{20 \cdot 20}
\][/tex]
3. Simplify the expression:
First, compute [tex]\(\sqrt{20}\)[/tex]. Remember that [tex]\(20 = 4 \times 5\)[/tex], so:
[tex]\[
\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}
\][/tex]
Now, substitute [tex]\(\sqrt{20}\)[/tex] back into the fraction:
[tex]\[
\frac{2\sqrt{5}}{20 \cdot 20}
\][/tex]
4. Simplify further:
Notice that [tex]\(20 \cdot 20 = 400\)[/tex]. So, we have:
[tex]\[
\frac{2\sqrt{5}}{400}
\][/tex]
5. Simplify the fraction:
Reduce the fraction by dividing the numerator and the denominator by 2:
[tex]\[
\frac{2\sqrt{5}}{400} = \frac{\sqrt{5}}{200}
\][/tex]
Thus, the fraction [tex]\(\frac{1}{20 \sqrt{20}}\)[/tex] can indeed be written as [tex]\(\frac{\sqrt{5}}{n}\)[/tex]. Comparing [tex]\(\frac{\sqrt{5}}{200}\)[/tex] with [tex]\(\frac{\sqrt{5}}{n}\)[/tex], we can see that:
[tex]\[
n = 200
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is [tex]\( 200 \)[/tex].
