Pre-Algebra Examples

$\frac{\sqrt{5}}{9 - \sqrt{65}}$

Step 1

Multiply $\frac{\sqrt{5}}{9 - \sqrt{65}}$ by $\frac{9 + \sqrt{65}}{9 + \sqrt{65}}$ .

$\frac{\sqrt{5}}{9 - \sqrt{65}} \cdot \frac{9 + \sqrt{65}}{9 + \sqrt{65}}$

Step 2

Multiply $\frac{\sqrt{5}}{9 - \sqrt{65}}$ by $\frac{9 + \sqrt{65}}{9 + \sqrt{65}}$ .

$\frac{\sqrt{5} (9 + \sqrt{65})}{(9 - \sqrt{65}) (9 + \sqrt{65})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} (9 + \sqrt{65})}{81 + 9 \sqrt{65} - 9 \sqrt{65} - {\sqrt{65}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{5} (9 + \sqrt{65})}{16}$

Step 5

Apply the distributive property.

$\frac{\sqrt{5} \cdot 9 + \sqrt{5} \sqrt{65}}{16}$

Step 6

Move $9$ to the left of $\sqrt{5}$ .

$\frac{9 \cdot \sqrt{5} + \sqrt{5} \sqrt{65}}{16}$

Step 7

Combine using the product rule for radicals.

$\frac{9 \cdot \sqrt{5} + \sqrt{5 \cdot 65}}{16}$

Step 8

Simplify each term.

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Step 8.1

Multiply $5$ by $65$ .

$\frac{9 \sqrt{5} + \sqrt{325}}{16}$

Step 8.2

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 8.2.1

Factor $25$ out of $325$ .

$\frac{9 \sqrt{5} + \sqrt{25 (13)}}{16}$

Step 8.2.2

Rewrite $25$ as $5^{2}$ .

$\frac{9 \sqrt{5} + \sqrt{5^{2} \cdot 13}}{16}$

Step 8.3

Pull terms out from under the radical.

$\frac{9 \sqrt{5} + 5 \sqrt{13}}{16}$

Step 9

Split the fraction $\frac{9 \sqrt{5} + 5 \sqrt{13}}{16}$ into two fractions.

$\frac{9 \sqrt{5}}{16} + \frac{5 \sqrt{13}}{16}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{9 \sqrt{5}}{16} + \frac{5 \sqrt{13}}{16}$

Decimal Form:

$2.38452301 \dots$