Trigonometry Examples

$\frac{4 \sqrt{2}}{5 - \sqrt{3}}$

Step 1

Multiply $\frac{4 \sqrt{2}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

$\frac{4 \sqrt{2}}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}}$

Step 2

Multiply $\frac{4 \sqrt{2}}{5 - \sqrt{3}}$ by $\frac{5 + \sqrt{3}}{5 + \sqrt{3}}$ .

$\frac{4 \sqrt{2} (5 + \sqrt{3})}{(5 - \sqrt{3}) (5 + \sqrt{3})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{4 \sqrt{2} (5 + \sqrt{3})}{25 + 5 \sqrt{3} - 5 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{4 \sqrt{2} (5 + \sqrt{3})}{22}$

Step 5

Cancel the common factor of $4$ and $22$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $4 \sqrt{2} (5 + \sqrt{3})$ .

$\frac{2 (2 \sqrt{2} (5 + \sqrt{3}))}{22}$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $2$ out of $22$ .

$\frac{2 (2 \sqrt{2} (5 + \sqrt{3}))}{2 (11)}$

Step 5.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{2} (5 + \sqrt{3}))}{2 \cdot 11}$

Step 5.2.3

Rewrite the expression.

$\frac{2 \sqrt{2} (5 + \sqrt{3})}{11}$

Step 6

Group $5 + \sqrt{3}$ and $\sqrt{2}$ together.

$\frac{2 ((5 + \sqrt{3}) \sqrt{2})}{11}$

Step 7

Apply the distributive property.

$\frac{2 (5 \sqrt{2} + \sqrt{3} \sqrt{2})}{11}$

Step 8

Combine using the product rule for radicals.

$\frac{2 (5 \sqrt{2} + \sqrt{3 \cdot 2})}{11}$

Step 9

Multiply $3$ by $2$ .

$\frac{2 (5 \sqrt{2} + \sqrt{6})}{11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2 (5 \sqrt{2} + \sqrt{6})}{11}$

Decimal Form:

$1.73101046 \dots$