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Problem 8

Number 8

\frac{(8 + \sqrt{5}) + (2\sqrt{5} + 1)}{\sqrt{5} + 3} = \dots

$2$

$3$

$5$

$10$

$14$

Explanation

Simplifying Algebraic Expression

We simplify the numerator by grouping like terms (integers with integers, radicals with radicals).

\begin{aligned} \frac{(8 + \sqrt{5}) + (2\sqrt{5} + 1)}{\sqrt{5} + 3} &= \frac{(8 + 1) + (\sqrt{5} + 2\sqrt{5})}{\sqrt{5} + 3} \\ &= \frac{9 + 3\sqrt{5}}{\sqrt{5} + 3} \end{aligned}

Next, we factor the numerator by taking out the number $3$ :

\begin{aligned} \frac{9 + 3\sqrt{5}}{\sqrt{5} + 3} &= \frac{3(3 + \sqrt{5})}{\sqrt{5} + 3} \end{aligned}

Notice that $(3 + \sqrt{5})$ is equal to $(\sqrt{5} + 3)$ . Thus, we can divide both the numerator and the denominator by this value:

\frac{3(3 + \sqrt{5})}{\sqrt{5} + 3} = 3

Therefore, the final result is $3$ .

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Problem 8Set 9

Exercises

Number 8

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