Question

8.5 divide radical expressions (homework)
score: 13/14 answered: 13/14
question 9
rationalize the denominator:
\\(\frac{\sqrt{2}}{7 - \sqrt{5}}=\\)
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Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $7 + \sqrt{5}$.
$\frac{\sqrt{2}(7+\sqrt{5})}{(7 - \sqrt{5})(7+\sqrt{5})}$

Step2: Expand numerator and denominator

Expand numerator: $\sqrt{2}(7+\sqrt{5})=7\sqrt{2}+\sqrt{10}$. Expand denominator using $(a - b)(a + b)=a^{2}-b^{2}$, so $(7 - \sqrt{5})(7+\sqrt{5})=7^{2}-(\sqrt{5})^{2}=49 - 5=44$.
$\frac{7\sqrt{2}+\sqrt{10}}{44}$

Answer:

$\frac{7\sqrt{2}+\sqrt{10}}{44}$