To simplify the expression, follow the steps below.
Step 01: Multiply the numerator and the denominator by "8 + √5".
[tex]\frac{5+\sqrt{5}}{8-\sqrt{5}}*\frac{8+\sqrt{5}}{8+\sqrt{5}}[/tex]
Step 02: Use the distributive property of multiplication.
According to the property:
[tex](a+b)*(c+d)=a*c+a*d+b*c+b*d[/tex]
Then,
[tex]\begin{gathered} \frac{5*8+5*\sqrt{5}+\sqrt{5}*8+\sqrt{5}*\sqrt{5}}{8*8+8*\sqrt{5}-\sqrt{5}*8-\sqrt{5}*\sqrt{5}} \\ \frac{40+5\sqrt{5}+8\sqrt{5}+\sqrt{25}}{64+8\sqrt{5}-8\sqrt{5}-\sqrt{25}} \end{gathered}[/tex]
Step 03: Sum the like-terms.
[tex]\begin{gathered} \frac{40+13\sqrt{5}+5}{64-5} \\ \frac{45+13\sqrt{5}}{59} \end{gathered}[/tex]
Answer:
[tex]\frac{45+13\sqrt{5}}{59}[/tex]
