Algebra Examples

$\frac{6 - \sqrt{14}}{2 + \sqrt{14}}$

Step 1

Multiply $\frac{6 - \sqrt{14}}{2 + \sqrt{14}}$ by $\frac{2 - \sqrt{14}}{2 - \sqrt{14}}$ .

$\frac{6 - \sqrt{14}}{2 + \sqrt{14}} \cdot \frac{2 - \sqrt{14}}{2 - \sqrt{14}}$

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Multiply $\frac{6 - \sqrt{14}}{2 + \sqrt{14}}$ by $\frac{2 - \sqrt{14}}{2 - \sqrt{14}}$ .

$\frac{(6 - \sqrt{14}) (2 - \sqrt{14})}{(2 + \sqrt{14}) (2 - \sqrt{14})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{14}) (2 - \sqrt{14})}{4 - 2 \sqrt{14} + \sqrt{14} \cdot 2 - {\sqrt{14}}^{2}}$

Step 2.3

Simplify.

$\frac{(6 - \sqrt{14}) (2 - \sqrt{14})}{- 10}$

Step 3

Expand $(6 - \sqrt{14}) (2 - \sqrt{14})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{6 (2 - \sqrt{14}) - \sqrt{14} (2 - \sqrt{14})}{- 10}$

Step 3.2

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{14}) - \sqrt{14} (2 - \sqrt{14})}{- 10}$

Step 3.3

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{14}) - \sqrt{14} \cdot 2 - \sqrt{14} (- \sqrt{14})}{- 10}$

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

Multiply $6$ by $2$ .

$\frac{12 + 6 (- \sqrt{14}) - \sqrt{14} \cdot 2 - \sqrt{14} (- \sqrt{14})}{- 10}$

Step 4.1.2

Multiply $- 1$ by $6$ .

$\frac{12 - 6 \sqrt{14} - \sqrt{14} \cdot 2 - \sqrt{14} (- \sqrt{14})}{- 10}$

Step 4.1.3

Multiply $2$ by $- 1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} - \sqrt{14} (- \sqrt{14})}{- 10}$

Step 4.1.4

Multiply $- \sqrt{14} (- \sqrt{14})$ .

Tap for more steps...

Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 1 \sqrt{14} \sqrt{14}}{- 10}$

Step 4.1.4.2

Multiply $\sqrt{14}$ by $1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + \sqrt{14} \sqrt{14}}{- 10}$

Step 4.1.4.3

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + {\sqrt{14}}^{1} \sqrt{14}}{- 10}$

Step 4.1.4.4

Raise $\sqrt{14}$ to the power of $1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + {\sqrt{14}}^{1} {\sqrt{14}}^{1}}{- 10}$

Step 4.1.4.5

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

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + {\sqrt{14}}^{1 + 1}}{- 10}$

Step 4.1.4.6

Add $1$ and $1$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + {\sqrt{14}}^{2}}{- 10}$

Step 4.1.5

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Tap for more steps...

Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + {(14^{\frac{1}{2}})}^{2}}{- 10}$

Step 4.1.5.2

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

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 14^{\frac{1}{2} \cdot 2}}{- 10}$

Step 4.1.5.3

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

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 14^{\frac{2}{2}}}{- 10}$

Step 4.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.5.4.1

Cancel the common factor.

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 14^{\frac{2}{2}}}{- 10}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 14^{1}}{- 10}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{12 - 6 \sqrt{14} - 2 \sqrt{14} + 14}{- 10}$

Step 4.2

Add $12$ and $14$ .

$\frac{26 - 6 \sqrt{14} - 2 \sqrt{14}}{- 10}$

Step 4.3

Subtract $2 \sqrt{14}$ from $- 6 \sqrt{14}$ .

$\frac{26 - 8 \sqrt{14}}{- 10}$

Step 5

Cancel the common factor of $26 - 8 \sqrt{14}$ and $- 10$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $26$ .

$\frac{2 (13) - 8 \sqrt{14}}{- 10}$

Step 5.2

Factor $2$ out of $- 8 \sqrt{14}$ .

$\frac{2 (13) + 2 (- 4 \sqrt{14})}{- 10}$

Step 5.3

Factor $2$ out of $2 (13) + 2 (- 4 \sqrt{14})$ .

$\frac{2 (13 - 4 \sqrt{14})}{- 10}$

Step 5.4

Cancel the common factors.

Tap for more steps...

Step 5.4.1

Factor $2$ out of $- 10$ .

$\frac{2 (13 - 4 \sqrt{14})}{2 \cdot - 5}$

Step 5.4.2

Cancel the common factor.

$\frac{2 (13 - 4 \sqrt{14})}{2 \cdot - 5}$

Step 5.4.3

Rewrite the expression.

$\frac{13 - 4 \sqrt{14}}{- 5}$

Step 6

Move the negative in front of the fraction.

$- \frac{13 - 4 \sqrt{14}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{13 - 4 \sqrt{14}}{5}$

Decimal Form:

$0.39332590 \dots$