Trigonometry Examples

$\frac{5 - \sqrt{7}}{9 - 9 \sqrt{2}}$

Step 1

Multiply $\frac{5 - \sqrt{7}}{9 - 9 \sqrt{2}}$ by $\frac{9 + 9 \sqrt{2}}{9 + 9 \sqrt{2}}$ .

$\frac{5 - \sqrt{7}}{9 - 9 \sqrt{2}} \cdot \frac{9 + 9 \sqrt{2}}{9 + 9 \sqrt{2}}$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Multiply $\frac{5 - \sqrt{7}}{9 - 9 \sqrt{2}}$ by $\frac{9 + 9 \sqrt{2}}{9 + 9 \sqrt{2}}$ .

$\frac{(5 - \sqrt{7}) (9 + 9 \sqrt{2})}{(9 - 9 \sqrt{2}) (9 + 9 \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 - \sqrt{7}) (9 + 9 \sqrt{2})}{81 + 81 \sqrt{2} - 81 \sqrt{2} - 81 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 - \sqrt{7}) (9 + 9 \sqrt{2})}{- 81}$

Step 2.4

Cancel the common factor of $9 + 9 \sqrt{2}$ and $- 81$ .

Tap for more steps...

Step 2.4.1

Factor $9$ out of $(5 - \sqrt{7}) (9 + 9 \sqrt{2})$ .

$\frac{9 ((5 - \sqrt{7}) (1 + \sqrt{2}))}{- 81}$

Step 2.4.2

Cancel the common factors.

Tap for more steps...

Step 2.4.2.1

Factor $9$ out of $- 81$ .

$\frac{9 ((5 - \sqrt{7}) (1 + \sqrt{2}))}{9 (- 9)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{9 ((5 - \sqrt{7}) (1 + \sqrt{2}))}{9 \cdot - 9}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(5 - \sqrt{7}) (1 + \sqrt{2})}{- 9}$

Step 3

Expand $(5 - \sqrt{7}) (1 + \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{5 (1 + \sqrt{2}) - \sqrt{7} (1 + \sqrt{2})}{- 9}$

Step 3.2

Apply the distributive property.

$\frac{5 \cdot 1 + 5 \sqrt{2} - \sqrt{7} (1 + \sqrt{2})}{- 9}$

Step 3.3

Apply the distributive property.

$\frac{5 \cdot 1 + 5 \sqrt{2} - \sqrt{7} \cdot 1 - \sqrt{7} \sqrt{2}}{- 9}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $5$ by $1$ .

$\frac{5 + 5 \sqrt{2} - \sqrt{7} \cdot 1 - \sqrt{7} \sqrt{2}}{- 9}$

Step 4.1.2

Multiply $- 1$ by $1$ .

$\frac{5 + 5 \sqrt{2} - \sqrt{7} - \sqrt{7} \sqrt{2}}{- 9}$

Step 4.1.3

Multiply $- \sqrt{7} \sqrt{2}$ .

Tap for more steps...

Step 4.1.3.1

Combine using the product rule for radicals.

$\frac{5 + 5 \sqrt{2} - \sqrt{7} - \sqrt{2 \cdot 7}}{- 9}$

Step 4.1.3.2

Multiply $2$ by $7$ .

$\frac{5 + 5 \sqrt{2} - \sqrt{7} - \sqrt{14}}{- 9}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{5 + 5 \sqrt{2} - \sqrt{7} - \sqrt{14}}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 + 5 \sqrt{2} - \sqrt{7} - \sqrt{14}}{9}$

Decimal Form:

$- 0.63151767 \dots$