Algebra Examples

$\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$

Step 1

Multiply $\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$ by $\frac{2 - \sqrt{7}}{2 - \sqrt{7}}$ .

$\frac{6 - \sqrt{3}}{2 + \sqrt{7}} \cdot \frac{2 - \sqrt{7}}{2 - \sqrt{7}}$

Step 2

Multiply $\frac{6 - \sqrt{3}}{2 + \sqrt{7}}$ by $\frac{2 - \sqrt{7}}{2 - \sqrt{7}}$ .

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{(2 + \sqrt{7}) (2 - \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{4 - 2 \sqrt{7} + \sqrt{7} \cdot 2 - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{(6 - \sqrt{3}) (2 - \sqrt{7})}{- 3}$

Step 5

Expand $(6 - \sqrt{3}) (2 - \sqrt{7})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{6 (2 - \sqrt{7}) - \sqrt{3} (2 - \sqrt{7})}{- 3}$

Step 5.2

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{7}) - \sqrt{3} (2 - \sqrt{7})}{- 3}$

Step 5.3

Apply the distributive property.

$\frac{6 \cdot 2 + 6 (- \sqrt{7}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Step 6

Simplify each term.

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Step 6.1

Multiply $6$ by $2$ .

$\frac{12 + 6 (- \sqrt{7}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Step 6.2

Multiply $- 1$ by $6$ .

$\frac{12 - 6 \sqrt{7} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{7})}{- 3}$

Step 6.3

Multiply $2$ by $- 1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{7})}{- 3}$

Step 6.4

Multiply $- \sqrt{3} (- \sqrt{7})$ .

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{7}}{- 3}$

Step 6.4.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{3} \sqrt{7}}{- 3}$

Step 6.4.3

Combine using the product rule for radicals.

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{3 \cdot 7}}{- 3}$

Step 6.4.4

Multiply $3$ by $7$ .

$\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{- 3}$

Step 7

Move the negative in front of the fraction.

$- \frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{3}$

Step 8

Split the fraction $\frac{12 - 6 \sqrt{7} - 2 \sqrt{3} + \sqrt{21}}{3}$ into two fractions.

$- (\frac{12 - 6 \sqrt{7} - 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 9

Split the fraction $\frac{12 - 6 \sqrt{7} - 2 \sqrt{3}}{3}$ into two fractions.

$- (\frac{12 - 6 \sqrt{7}}{3} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 10

Split the fraction $\frac{12 - 6 \sqrt{7}}{3}$ into two fractions.

$- (\frac{12}{3} + \frac{- 6 \sqrt{7}}{3} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 11

Divide $12$ by $3$ .

$- (4 + \frac{- 6 \sqrt{7}}{3} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 12

Cancel the common factor of $- 6$ and $3$ .

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Step 12.1

Factor $3$ out of $- 6 \sqrt{7}$ .

$- (4 + \frac{3 (- 2 \sqrt{7})}{3} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $3$ out of $3$ .

$- (4 + \frac{3 (- 2 \sqrt{7})}{3 (1)} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 12.2.2

Cancel the common factor.

$- (4 + \frac{3 (- 2 \sqrt{7})}{3 \cdot 1} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 12.2.3

Rewrite the expression.

$- (4 + \frac{- 2 \sqrt{7}}{1} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 12.2.4

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

$- (4 - 2 \sqrt{7} + \frac{- 2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 13

Move the negative in front of the fraction.

$- (4 - 2 \sqrt{7} - \frac{2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Step 14

The result can be shown in multiple forms.

Exact Form:

$- (4 - 2 \sqrt{7} - \frac{2 \sqrt{3}}{3} + \frac{\sqrt{21}}{3})$

Decimal Form:

$0.91867792 \dots$