To rationalize the numerator of the fraction [tex]\(\sqrt{\frac{13}{19}}\)[/tex], follow these steps:
1. Identify the given fraction under the square root: [tex]\(\sqrt{\frac{13}{19}}\)[/tex]
2. Simplify by considering the numerator and denominator separately under the square root:
[tex]\[
\sqrt{\frac{13}{19}} = \frac{\sqrt{13}}{\sqrt{19}}
\][/tex]
3. Calculate the square root of the numerator ([tex]\(\sqrt{13}\)[/tex]) and the square root of the denominator ([tex]\(\sqrt{19}\)[/tex]) respectively:
[tex]\[
\sqrt{13} \approx 3.605551275463989
\][/tex]
[tex]\[
\sqrt{19} \approx 4.358898943540674
\][/tex]
4. Rationalize the numerator by multiplying the fraction by [tex]\(\frac{\sqrt{19}}{\sqrt{19}}\)[/tex] (this is effectively multiplying by 1, to adjust the expression):
[tex]\[
\frac{\sqrt{13}}{\sqrt{19}} \times \frac{\sqrt{19}}{\sqrt{19}} = \frac{\sqrt{13} \times \sqrt{19}}{19}
\][/tex]
5. Calculate the product of the square roots in the numerator:
[tex]\[
\sqrt{13} \times \sqrt{19} \approx 3.605551275463989 \times 4.358898943540674 \approx 15.716233645501712
\][/tex]
6. The denominator is the square of [tex]\(\sqrt{19}\)[/tex], which simplifies to 19:
[tex]\[
\sqrt{19} \times \sqrt{19} = 19
\][/tex]
7. Combine the results to state the rationalized form:
[tex]\[
\frac{\sqrt{13} \times \sqrt{19}}{19} \approx \frac{15.716233645501712}{19}
\][/tex]
Therefore, the rationalized form of [tex]\(\sqrt{\frac{13}{19}}\)[/tex] is [tex]\(\frac{15.716233645501712}{19}\)[/tex]. The individual components involved in reaching this solution are:
- [tex]\(\sqrt{13} \approx 3.605551275463989\)[/tex]
- [tex]\(\sqrt{19} \approx 4.358898943540674\)[/tex]
- Numerator after rationalizing: [tex]\(15.716233645501712\)[/tex]
- Denominator: [tex]\(19\)[/tex]
So the fully rationalized expression results in the fraction:
[tex]\[
\frac{15.716233645501712}{19}
\][/tex]
