Pre-Algebra Examples

$\frac{8 - 9 \sqrt{3}}{2 - \sqrt{6}}$

Step 1

Multiply $\frac{8 - 9 \sqrt{3}}{2 - \sqrt{6}}$ by $\frac{2 + \sqrt{6}}{2 + \sqrt{6}}$ .

$\frac{8 - 9 \sqrt{3}}{2 - \sqrt{6}} \cdot \frac{2 + \sqrt{6}}{2 + \sqrt{6}}$

Step 2

Multiply $\frac{8 - 9 \sqrt{3}}{2 - \sqrt{6}}$ by $\frac{2 + \sqrt{6}}{2 + \sqrt{6}}$ .

$\frac{(8 - 9 \sqrt{3}) (2 + \sqrt{6})}{(2 - \sqrt{6}) (2 + \sqrt{6})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(8 - 9 \sqrt{3}) (2 + \sqrt{6})}{4 + 2 \sqrt{6} - 2 \sqrt{6} - {\sqrt{6}}^{2}}$

Step 4

Simplify.

$\frac{(8 - 9 \sqrt{3}) (2 + \sqrt{6})}{- 2}$

Step 5

Expand $(8 - 9 \sqrt{3}) (2 + \sqrt{6})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{8 (2 + \sqrt{6}) - 9 \sqrt{3} (2 + \sqrt{6})}{- 2}$

Step 5.2

Apply the distributive property.

$\frac{8 \cdot 2 + 8 \sqrt{6} - 9 \sqrt{3} (2 + \sqrt{6})}{- 2}$

Step 5.3

Apply the distributive property.

$\frac{8 \cdot 2 + 8 \sqrt{6} - 9 \sqrt{3} \cdot 2 - 9 \sqrt{3} \sqrt{6}}{- 2}$

Step 6

Simplify each term.

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Step 6.1

Multiply $8$ by $2$ .

$\frac{16 + 8 \sqrt{6} - 9 \sqrt{3} \cdot 2 - 9 \sqrt{3} \sqrt{6}}{- 2}$

Step 6.2

Multiply $2$ by $- 9$ .

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 \sqrt{3} \sqrt{6}}{- 2}$

Step 6.3

Multiply $- 9 \sqrt{3} \sqrt{6}$ .

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Step 6.3.1

Combine using the product rule for radicals.

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 \sqrt{6 \cdot 3}}{- 2}$

Step 6.3.2

Multiply $6$ by $3$ .

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 \sqrt{18}}{- 2}$

Step 6.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.4.1

Factor $9$ out of $18$ .

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 \sqrt{9 (2)}}{- 2}$

Step 6.4.2

Rewrite $9$ as $3^{2}$ .

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 \sqrt{3^{2} \cdot 2}}{- 2}$

Step 6.5

Pull terms out from under the radical.

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 9 (3 \sqrt{2})}{- 2}$

Step 6.6

Multiply $3$ by $- 9$ .

$\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 27 \sqrt{2}}{- 2}$

Step 7

Move the negative in front of the fraction.

$- \frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 27 \sqrt{2}}{2}$

Step 8

Split the fraction $\frac{16 + 8 \sqrt{6} - 18 \sqrt{3} - 27 \sqrt{2}}{2}$ into two fractions.

$- (\frac{16 + 8 \sqrt{6} - 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 9

Split the fraction $\frac{16 + 8 \sqrt{6} - 18 \sqrt{3}}{2}$ into two fractions.

$- (\frac{16 + 8 \sqrt{6}}{2} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 10

Split the fraction $\frac{16 + 8 \sqrt{6}}{2}$ into two fractions.

$- (\frac{16}{2} + \frac{8 \sqrt{6}}{2} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 11

Divide $16$ by $2$ .

$- (8 + \frac{8 \sqrt{6}}{2} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 12

Cancel the common factor of $8$ and $2$ .

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Step 12.1

Factor $2$ out of $8 \sqrt{6}$ .

$- (8 + \frac{2 (4 \sqrt{6})}{2} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $2$ out of $2$ .

$- (8 + \frac{2 (4 \sqrt{6})}{2 (1)} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 12.2.2

Cancel the common factor.

$- (8 + \frac{2 (4 \sqrt{6})}{2 \cdot 1} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 12.2.3

Rewrite the expression.

$- (8 + \frac{4 \sqrt{6}}{1} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 12.2.4

Divide $4 \sqrt{6}$ by $1$ .

$- (8 + 4 \sqrt{6} + \frac{- 18 \sqrt{3}}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 13

Cancel the common factor of $- 18$ and $2$ .

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Step 13.1

Factor $2$ out of $- 18 \sqrt{3}$ .

$- (8 + 4 \sqrt{6} + \frac{2 (- 9 \sqrt{3})}{2} + \frac{- 27 \sqrt{2}}{2})$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $2$ out of $2$ .

$- (8 + 4 \sqrt{6} + \frac{2 (- 9 \sqrt{3})}{2 (1)} + \frac{- 27 \sqrt{2}}{2})$

Step 13.2.2

Cancel the common factor.

$- (8 + 4 \sqrt{6} + \frac{2 (- 9 \sqrt{3})}{2 \cdot 1} + \frac{- 27 \sqrt{2}}{2})$

Step 13.2.3

Rewrite the expression.

$- (8 + 4 \sqrt{6} + \frac{- 9 \sqrt{3}}{1} + \frac{- 27 \sqrt{2}}{2})$

Step 13.2.4

Divide $- 9 \sqrt{3}$ by $1$ .

$- (8 + 4 \sqrt{6} - 9 \sqrt{3} + \frac{- 27 \sqrt{2}}{2})$

Step 14

Move the negative in front of the fraction.

$- (8 + 4 \sqrt{6} - 9 \sqrt{3} - \frac{27 \sqrt{2}}{2})$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- (8 + 4 \sqrt{6} - 9 \sqrt{3} - \frac{27 \sqrt{2}}{2})$

Decimal Form:

$16.88238138 \dots$