Trigonometry Examples

$\frac{\sqrt{10} - 8}{4 + \sqrt{10}}$

Step 1

Multiply $\frac{\sqrt{10} - 8}{4 + \sqrt{10}}$ by $\frac{4 - \sqrt{10}}{4 - \sqrt{10}}$ .

$\frac{\sqrt{10} - 8}{4 + \sqrt{10}} \cdot \frac{4 - \sqrt{10}}{4 - \sqrt{10}}$

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Multiply $\frac{\sqrt{10} - 8}{4 + \sqrt{10}}$ by $\frac{4 - \sqrt{10}}{4 - \sqrt{10}}$ .

$\frac{(\sqrt{10} - 8) (4 - \sqrt{10})}{(4 + \sqrt{10}) (4 - \sqrt{10})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{10} - 8) (4 - \sqrt{10})}{16 - 4 \sqrt{10} + \sqrt{10} \cdot 4 - {\sqrt{10}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{10} - 8) (4 - \sqrt{10})}{6}$

Step 3

Expand $(\sqrt{10} - 8) (4 - \sqrt{10})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{\sqrt{10} (4 - \sqrt{10}) - 8 (4 - \sqrt{10})}{6}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{10} \cdot 4 + \sqrt{10} (- \sqrt{10}) - 8 (4 - \sqrt{10})}{6}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{10} \cdot 4 + \sqrt{10} (- \sqrt{10}) - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Move $4$ to the left of $\sqrt{10}$ .

$\frac{4 \cdot \sqrt{10} + \sqrt{10} (- \sqrt{10}) - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.2

Multiply $\sqrt{10} (- \sqrt{10})$ .

Tap for more steps...

Step 4.1.2.1

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{4 \sqrt{10} - ({\sqrt{10}}^{1} \sqrt{10}) - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.2.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{4 \sqrt{10} - ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.2.3

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

$\frac{4 \sqrt{10} - {\sqrt{10}}^{1 + 1} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{4 \sqrt{10} - {\sqrt{10}}^{2} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Tap for more steps...

Step 4.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{4 \sqrt{10} - {(10^{\frac{1}{2}})}^{2} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3.2

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

$\frac{4 \sqrt{10} - 10^{\frac{1}{2} \cdot 2} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3.3

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

$\frac{4 \sqrt{10} - 10^{\frac{2}{2}} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.3.4.1

Cancel the common factor.

$\frac{4 \sqrt{10} - 10^{\frac{2}{2}} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{4 \sqrt{10} - 10^{1} - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{4 \sqrt{10} - 1 \cdot 10 - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.4

Multiply $- 1$ by $10$ .

$\frac{4 \sqrt{10} - 10 - 8 \cdot 4 - 8 (- \sqrt{10})}{6}$

Step 4.1.5

Multiply $- 8$ by $4$ .

$\frac{4 \sqrt{10} - 10 - 32 - 8 (- \sqrt{10})}{6}$

Step 4.1.6

Multiply $- 1$ by $- 8$ .

$\frac{4 \sqrt{10} - 10 - 32 + 8 \sqrt{10}}{6}$

Step 4.2

Add $4 \sqrt{10}$ and $8 \sqrt{10}$ .

$\frac{- 10 - 32 + 12 \sqrt{10}}{6}$

Step 4.3

Subtract $32$ from $- 10$ .

$\frac{- 42 + 12 \sqrt{10}}{6}$

Step 5

Cancel the common factor of $- 42 + 12 \sqrt{10}$ and $6$ .

Tap for more steps...

Step 5.1

Factor $6$ out of $- 42$ .

$\frac{6 \cdot - 7 + 12 \sqrt{10}}{6}$

Step 5.2

Factor $6$ out of $12 \sqrt{10}$ .

$\frac{6 \cdot - 7 + 6 (2 \sqrt{10})}{6}$

Step 5.3

Factor $6$ out of $6 (- 7) + 6 (2 \sqrt{10})$ .

$\frac{6 (- 7 + 2 \sqrt{10})}{6}$

Step 5.4

Cancel the common factors.

Tap for more steps...

Step 5.4.1

Factor $6$ out of $6$ .

$\frac{6 (- 7 + 2 \sqrt{10})}{6 (1)}$

Step 5.4.2

Cancel the common factor.

$\frac{6 (- 7 + 2 \sqrt{10})}{6 \cdot 1}$

Step 5.4.3

Rewrite the expression.

$\frac{- 7 + 2 \sqrt{10}}{1}$

Step 5.4.4

Divide $- 7 + 2 \sqrt{10}$ by $1$ .

$- 7 + 2 \sqrt{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 7 + 2 \sqrt{10}$

Decimal Form:

$- 0.67544467 \dots$