Question

8.5 divide radical expressions (homework)
score: 11/14 answered: 12/14
question 13
rationalize the denominator:
\\(\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}} =\\)
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Explanation:

Step1: Multiply by conjugate

Multiply the fraction $\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}}$ by $\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}$.
\[

$$\begin{align*} \frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}+\sqrt{5}}\times\frac{\sqrt{6}-\sqrt{5}}{\sqrt{6}-\sqrt{5}}&=\frac{(\sqrt{6}-\sqrt{5})^2}{(\sqrt{6}+\sqrt{5})(\sqrt{6}-\sqrt{5})} \end{align*}$$

\]

Step2: Expand numerator and denominator

Expand the numerator $(\sqrt{6}-\sqrt{5})^2$ using $(a - b)^2=a^2-2ab + b^2$ and the denominator using $(a + b)(a - b)=a^2 - b^2$.
The numerator $(\sqrt{6}-\sqrt{5})^2=(\sqrt{6})^2-2\sqrt{6}\times\sqrt{5}+(\sqrt{5})^2=6 - 2\sqrt{30}+5=11-2\sqrt{30}$.
The denominator $(\sqrt{6}+\sqrt{5})(\sqrt{6}-\sqrt{5})=(\sqrt{6})^2-(\sqrt{5})^2=6 - 5 = 1$.

Step3: Simplify the fraction

Since the denominator is 1, the fraction simplifies to the numerator value.
The result is $11-2\sqrt{30}$.

Answer:

$11 - 2\sqrt{30}$