Pre-Algebra Examples

$\frac{5}{3 + \sqrt{5}}$

Step 1

Multiply $\frac{5}{3 + \sqrt{5}}$ by $\frac{3 - \sqrt{5}}{3 - \sqrt{5}}$ .

$\frac{5}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

Step 2

Multiply $\frac{5}{3 + \sqrt{5}}$ by $\frac{3 - \sqrt{5}}{3 - \sqrt{5}}$ .

$\frac{5 (3 - \sqrt{5})}{(3 + \sqrt{5}) (3 - \sqrt{5})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{5 (3 - \sqrt{5})}{9 - 3 \sqrt{5} + \sqrt{5} \cdot 3 - {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{5 (3 - \sqrt{5})}{4}$

Step 5

Apply the distributive property.

$\frac{5 \cdot 3 + 5 (- \sqrt{5})}{4}$

Step 6

Multiply $5$ by $3$ .

$\frac{15 + 5 (- \sqrt{5})}{4}$

Step 7

Multiply $- 1$ by $5$ .

$\frac{15 - 5 \sqrt{5}}{4}$

Step 8

Split the fraction $\frac{15 - 5 \sqrt{5}}{4}$ into two fractions.

$\frac{15}{4} + \frac{- 5 \sqrt{5}}{4}$

Step 9

Move the negative in front of the fraction.

$\frac{15}{4} - \frac{5 \sqrt{5}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{4} - \frac{5 \sqrt{5}}{4}$

Decimal Form:

$0.95491502 \dots$