To solve for the quotient
[tex]$
\frac{2}{\sqrt{13}+\sqrt{11}},
$[/tex]
we will rationalize the denominator, which involves a series of steps to eliminate the square roots in the denominator.
Step 1: Identify the initial fraction:
[tex]$
\frac{2}{\sqrt{13}+\sqrt{11}}.
$[/tex]
Step 2: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. The conjugate of [tex]\(\sqrt{13}+\sqrt{11}\)[/tex] is [tex]\(\sqrt{13}-\sqrt{11}\)[/tex]. Thus, we multiply by:
[tex]$
\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}.
$[/tex]
Step 3: Write the expression after multiplication:
[tex]$
\frac{2 \cdot (\sqrt{13}-\sqrt{11})}{(\sqrt{13}+\sqrt{11}) \cdot (\sqrt{13}-\sqrt{11})}.
$[/tex]
Step 4: Simplify the denominator using the difference of squares formula, [tex]\((a+b)(a-b) = a^2 - b^2\)[/tex]:
[tex]$
(\sqrt{13})^2 - (\sqrt{11})^2 = 13 - 11 = 2.
$[/tex]
Step 5: The numerator simplifies to:
[tex]$
2 \cdot (\sqrt{13} - \sqrt{11}).
$[/tex]
Step 6: Incorporate the simplified denominator:
[tex]$
\frac{2 (\sqrt{13} - \sqrt{11})}{2}.
$[/tex]
Step 7: Simplify the fraction by canceling out the common factor of 2 in the numerator and denominator:
[tex]$
\sqrt{13} - \sqrt{11}.
$[/tex]
Thus, the quotient is
[tex]$
\sqrt{13} - \sqrt{11}.
$[/tex]
