Question

8.5 divide radical expressions (homework)
score: 6/14 answered: 6/14
question 7
rationalize the denominator:
\\(\frac{2}{5 - \sqrt{7}}=\\)
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Explanation:

Step1: Multiply by conjugate

Multiply the fraction $\frac{2}{5 - \sqrt{7}}$ by $\frac{5+\sqrt{7}}{5+\sqrt{7}}$. So we get $\frac{2(5 + \sqrt{7})}{(5-\sqrt{7})(5+\sqrt{7})}$.

Step2: Expand denominator

Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a = 5$ and $b=\sqrt{7}$, so $(5-\sqrt{7})(5+\sqrt{7})=5^{2}-(\sqrt{7})^{2}=25 - 7=18$.

Step3: Expand numerator

Expand $2(5+\sqrt{7})$ to get $10 + 2\sqrt{7}$.

Step4: Simplify fraction

The fraction becomes $\frac{10 + 2\sqrt{7}}{18}=\frac{5+\sqrt{7}}{9}$.

Answer:

$\frac{5+\sqrt{7}}{9}$