Question

8.5 divide radical expressions (homework)
score: 12/14 answered: 13/14
question 14
rationalize the denominator. simplify your answer as much as possible.
\\(\frac{3}{\sqrt{a + h}-\sqrt{a}}=)

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $\sqrt{a + h}+\sqrt{a}$: $\frac{3(\sqrt{a + h}+\sqrt{a})}{(\sqrt{a + h}-\sqrt{a})(\sqrt{a + h}+\sqrt{a})}$

Step2: Expand denominator

Use difference - of - squares $(x - y)(x + y)=x^{2}-y^{2}$. Here $x=\sqrt{a + h}$ and $y = \sqrt{a}$, so denominator is $(\sqrt{a + h})^{2}-(\sqrt{a})^{2}=a + h - a=h$.

Step3: Simplify expression

The expression becomes $\frac{3(\sqrt{a + h}+\sqrt{a})}{h}=\frac{3\sqrt{a + h}+3\sqrt{a}}{h}$

Answer:

$\frac{3\sqrt{a + h}+3\sqrt{a}}{h}$