Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{\sqrt{5} (9 + \sqrt{5})}{76}$

Step 5

Apply the distributive property.

$\frac{\sqrt{5} \cdot 9 + \sqrt{5} \sqrt{5}}{76}$

Step 6

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

$\frac{9 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}}{76}$

Step 7

Combine using the product rule for radicals.

$\frac{9 \cdot \sqrt{5} + \sqrt{5 \cdot 5}}{76}$

Step 8

Simplify each term.

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Step 8.1

Multiply $5$ by $5$ .

$\frac{9 \sqrt{5} + \sqrt{25}}{76}$

Step 8.2

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

$\frac{9 \sqrt{5} + \sqrt{5^{2}}}{76}$

Step 8.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{9 \sqrt{5} + 5}{76}$

Step 9

Split the fraction $\frac{9 \sqrt{5} + 5}{76}$ into two fractions.

$\frac{9 \sqrt{5}}{76} + \frac{5}{76}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{9 \sqrt{5}}{76} + \frac{5}{76}$

Decimal Form:

$0.33058699 \dots$