Algebra Examples

$\frac{1}{\sqrt{5} + \sqrt{3} - 1}$

Step 1

Multiply $\frac{1}{\sqrt{5} + \sqrt{3} - 1}$ by $\frac{\sqrt{5} + \sqrt{3} + 1}{\sqrt{5} + \sqrt{3} + 1}$ .

$\frac{1}{\sqrt{5} + \sqrt{3} - 1} \cdot \frac{\sqrt{5} + \sqrt{3} + 1}{\sqrt{5} + \sqrt{3} + 1}$

Step 2

Multiply $\frac{1}{\sqrt{5} + \sqrt{3} - 1}$ by $\frac{\sqrt{5} + \sqrt{3} + 1}{\sqrt{5} + \sqrt{3} + 1}$ .

$\frac{\sqrt{5} + \sqrt{3} + 1}{(\sqrt{5} + \sqrt{3} - 1) (\sqrt{5} + \sqrt{3} + 1)}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} + \sqrt{3} + 1}{{\sqrt{5}}^{2} + \sqrt{15} + \sqrt{5} + \sqrt{15} + {\sqrt{3}}^{2} + \sqrt{3} - \sqrt{5} - \sqrt{3} - 1}$

Step 4

Simplify.

$\frac{\sqrt{5} + \sqrt{3} + 1}{7 + 2 \sqrt{15}}$

Step 5

Multiply $\frac{\sqrt{5} + \sqrt{3} + 1}{7 + 2 \sqrt{15}}$ by $\frac{7 - 2 \sqrt{15}}{7 - 2 \sqrt{15}}$ .

$\frac{\sqrt{5} + \sqrt{3} + 1}{7 + 2 \sqrt{15}} \cdot \frac{7 - 2 \sqrt{15}}{7 - 2 \sqrt{15}}$

Step 6

Multiply $\frac{\sqrt{5} + \sqrt{3} + 1}{7 + 2 \sqrt{15}}$ by $\frac{7 - 2 \sqrt{15}}{7 - 2 \sqrt{15}}$ .

$\frac{(\sqrt{5} + \sqrt{3} + 1) (7 - 2 \sqrt{15})}{(7 + 2 \sqrt{15}) (7 - 2 \sqrt{15})}$

Step 7

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} + \sqrt{3} + 1) (7 - 2 \sqrt{15})}{49 - 14 \sqrt{15} + 14 \sqrt{15} - 4 {\sqrt{15}}^{2}}$

Step 8

Simplify.

$\frac{(\sqrt{5} + \sqrt{3} + 1) (7 - 2 \sqrt{15})}{- 11}$

Step 9

Move the negative in front of the fraction.

$- \frac{(\sqrt{5} + \sqrt{3} + 1) (7 - 2 \sqrt{15})}{11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{(\sqrt{5} + \sqrt{3} + 1) (7 - 2 \sqrt{15})}{11}$

Decimal Form:

$0.33691373 \dots$