## Intermediate Algebra for College Students (7th Edition)

$\displaystyle \frac{a-b}{a-2\sqrt{ab}+b}$
We lose the square roots in the numerator by applying the difference of squares formula: $(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}$ $=a^{2}x-b^{2}y$ $\displaystyle \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\color{red}{ \cdot\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}} }\qquad$ (rationalize) $=\displaystyle \frac{(\sqrt{a})^{2}-(\sqrt{b})^{2}}{(\sqrt{a}-\sqrt{b})(\sqrt{a}-\sqrt{b})}$ ... the denominator is a square of a difference, $(A-B)^{2}=A^{2}-2AB+B^{2}$ $=\displaystyle \frac{a-b}{(\sqrt{a})^{2}-2\sqrt{ab}+(\sqrt{b})^{2}}$ $=\displaystyle \frac{a-b}{a-2\sqrt{ab}+b}$