Trigonometry Examples

$\sqrt{\frac{1 + \frac{2 \sqrt{2}}{3}}{1 - \frac{2 \sqrt{2}}{3}}}$

Step 1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{3}{3} + \frac{2 \sqrt{2}}{3}}{1 - \frac{2 \sqrt{2}}{3}}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{3 + 2 \sqrt{2}}{3}}{1 - \frac{2 \sqrt{2}}{3}}}$

Step 3

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{3 + 2 \sqrt{2}}{3}}{\frac{3}{3} - \frac{2 \sqrt{2}}{3}}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{3 + 2 \sqrt{2}}{3}}{\frac{3 - 2 \sqrt{2}}{3}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

$\sqrt{(3 + 2 \sqrt{2}) (\frac{1}{3 - 2 \sqrt{2}} \cdot \frac{3 + 2 \sqrt{2}}{3 + 2 \sqrt{2}})}$

Step 8

Combine fractions.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Expand the denominator using the FOIL method.

$\sqrt{(3 + 2 \sqrt{2}) \frac{3 + 2 \sqrt{2}}{9 + 6 \sqrt{2} - 6 \sqrt{2} - 4 {\sqrt{2}}^{2}}}$

Step 8.3

Simplify.

$\sqrt{(3 + 2 \sqrt{2}) \frac{3 + 2 \sqrt{2}}{1}}$

Step 8.4

Divide $3 + 2 \sqrt{2}$ by $1$ .

$\sqrt{(3 + 2 \sqrt{2}) (3 + 2 \sqrt{2})}$

Step 9

Expand $(3 + 2 \sqrt{2}) (3 + 2 \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\sqrt{3 (3 + 2 \sqrt{2}) + 2 \sqrt{2} (3 + 2 \sqrt{2})}$

Step 9.2

Apply the distributive property.

$\sqrt{3 \cdot 3 + 3 (2 \sqrt{2}) + 2 \sqrt{2} (3 + 2 \sqrt{2})}$

Step 9.3

Apply the distributive property.

$\sqrt{3 \cdot 3 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})}$

Step 10

Simplify and combine like terms.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Multiply $3$ by $3$ .

$\sqrt{9 + 3 (2 \sqrt{2}) + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})}$

Step 10.1.2

Multiply $2$ by $3$ .

$\sqrt{9 + 6 \sqrt{2} + 2 \sqrt{2} \cdot 3 + 2 \sqrt{2} (2 \sqrt{2})}$

Step 10.1.3

Multiply $3$ by $2$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 2 \sqrt{2} (2 \sqrt{2})}$

Step 10.1.4

Multiply $2 \sqrt{2} (2 \sqrt{2})$ .

Tap for more steps...

Step 10.1.4.1

Multiply $2$ by $2$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \sqrt{2} \sqrt{2}}$

Step 10.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 10.1.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 10.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {\sqrt{2}}^{1 + 1}}$

Step 10.1.4.5

Add $1$ and $1$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {\sqrt{2}}^{2}}$

Step 10.1.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 {(2^{\frac{1}{2}})}^{2}}$

Step 10.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}}$

Step 10.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1.5.4.1

Cancel the common factor.

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{\frac{2}{2}}}$

Step 10.1.5.4.2

Rewrite the expression.

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2^{1}}$

Step 10.1.5.5

Evaluate the exponent.

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 4 \cdot 2}$

Step 10.1.6

Multiply $4$ by $2$ .

$\sqrt{9 + 6 \sqrt{2} + 6 \sqrt{2} + 8}$

Step 10.2

Add $9$ and $8$ .

$\sqrt{17 + 6 \sqrt{2} + 6 \sqrt{2}}$

Step 10.3

Add $6 \sqrt{2}$ and $6 \sqrt{2}$ .

$\sqrt{17 + 12 \sqrt{2}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sqrt{17 + 12 \sqrt{2}}$

Decimal Form:

$5.82842712 \dots$