To determine which expression is equal to [tex]\(\frac{5 \sqrt{2}}{\sqrt{13}}\)[/tex], we start by simplifying the given expression:
[tex]\[
\frac{5 \sqrt{2}}{\sqrt{13}}
\][/tex]
One technique to simplify such expressions is to rationalize the denominator. However, it’s often easier to multiply both the numerator and the denominator by the square root present in the denominator to eliminate the square root from the denominator, while keeping the expression equal in value:
Multiplying both the numerator and denominator by [tex]\(\sqrt{13}\)[/tex]:
[tex]\[
\frac{5 \sqrt{2}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}} = \frac{5 \sqrt{2} \cdot \sqrt{13}}{\sqrt{13} \cdot \sqrt{13}}
\][/tex]
Recall that [tex]\(\sqrt{a} \cdot \sqrt{a} = a\)[/tex]. Therefore, [tex]\(\sqrt{13} \cdot \sqrt{13} = 13\)[/tex]:
[tex]\[
= \frac{5 \sqrt{2 \cdot 13}}{13}
= \frac{5 \sqrt{26}}{13}
\][/tex]
We have expressed [tex]\(\frac{5 \sqrt{2}}{\sqrt{13}}\)[/tex] in a simplified, rationalized form:
[tex]\[
\frac{5 \sqrt{26}}{13}
\][/tex]
Next, we compare this result to the provided options:
[tex]\[
1) \frac{5 \sqrt{26}}{13}
\][/tex]
[tex]\[
2) \frac{\sqrt{26}}{13}
\][/tex]
[tex]\[
3) \frac{5 \sqrt{26}}{\sqrt{13}}
\][/tex]
[tex]\[
4) \frac{\sqrt{65}}{2}
\][/tex]
The correct match is:
[tex]\[
\boxed{\frac{5 \sqrt{26}}{13}}
\][/tex]
