Intermediate Algebra for College Students (7th Edition)

$\displaystyle \frac{x-16}{x-4\sqrt{x}}$
We lose the square roots in the numerator by applying the difference of squares formula: $(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}$ $=a^{2}x-b^{2}y$ $\displaystyle \frac{\sqrt{x}+4}{\sqrt{x}}\color{red}{ \cdot\frac{\sqrt{x}-4}{\sqrt{x}-4} }\qquad$ (rationalize) $=\displaystyle \frac{(\sqrt{x})^{2}-4^{2}}{\sqrt{x}(\sqrt{x}-4)}$ $=\displaystyle \frac{x-16}{(\sqrt{x})^{2}-4\sqrt{x}}$ $=\displaystyle \frac{x-16}{x-4\sqrt{x}}$