To rationalize the denominator of the expression [tex]\(\frac{14n}{\sqrt{20n^3}}\)[/tex], follow these steps:
1. Rewrite the original expression:
[tex]\[
\frac{14n}{\sqrt{20n^3}}
\][/tex]
2. Simplify the expression inside the square root:
Notice that [tex]\(20n^3\)[/tex] can be broken down into its prime factors and powers of [tex]\(n\)[/tex]:
[tex]\[
20 = 4 \times 5 = 2^2 \times 5
\][/tex]
Hence,
[tex]\[
\sqrt{20n^3} = \sqrt{(2^2 \times 5 \times n^3)}
\][/tex]
3. Simplify the square root by separating perfect squares:
[tex]\[
\sqrt{2^2 \times 5 \times n^3} = \sqrt{(2^2 \times n^2) \times (5 \times n)} = 2n\sqrt{5n}
\][/tex]
4. Rewrite the original expression using this simplified denominator:
[tex]\[
\frac{14n}{2n\sqrt{5n}}
\][/tex]
5. Simplify the fraction by canceling common factors:
[tex]\[
\frac{14n}{2n\sqrt{5n}} = \frac{7}{\sqrt{5n}}
\][/tex]
6. Rationalize the denominator by multiplying the numerator and the denominator by [tex]\(\sqrt{5n}\)[/tex]:
[tex]\[
\frac{7}{\sqrt{5n}} \times \frac{\sqrt{5n}}{\sqrt{5n}} = \frac{7\sqrt{5n}}{5n}
\][/tex]
Thus, the simplified expression with the rationalized denominator is:
[tex]\[
\frac{7\sqrt{5n}}{5n}
\][/tex]
This is fully simplified, and the denominator is rationalized.
