Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(9 - 2 \sqrt{3}) (\sqrt{3} - 6)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 6 + 6 \sqrt{3} - 36}$

Step 2.3

Simplify.

$\frac{(9 - 2 \sqrt{3}) (\sqrt{3} - 6)}{- 33}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{9 (\sqrt{3} - 6) - 2 \sqrt{3} (\sqrt{3} - 6)}{- 33}$

Step 3.2

Apply the distributive property.

$\frac{9 \sqrt{3} + 9 \cdot - 6 - 2 \sqrt{3} (\sqrt{3} - 6)}{- 33}$

Step 3.3

Apply the distributive property.

$\frac{9 \sqrt{3} + 9 \cdot - 6 - 2 \sqrt{3} \sqrt{3} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $9$ by $- 6$ .

$\frac{9 \sqrt{3} - 54 - 2 \sqrt{3} \sqrt{3} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.2

Multiply $- 2 \sqrt{3} \sqrt{3}$ .

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Step 4.1.2.1

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{9 \sqrt{3} - 54 - 2 ({\sqrt{3}}^{1} \sqrt{3}) - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{9 \sqrt{3} - 54 - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.2.3

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

$\frac{9 \sqrt{3} - 54 - 2 {\sqrt{3}}^{1 + 1} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{9 \sqrt{3} - 54 - 2 {\sqrt{3}}^{2} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{9 \sqrt{3} - 54 - 2 {(3^{\frac{1}{2}})}^{2} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3.2

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

$\frac{9 \sqrt{3} - 54 - 2 \cdot 3^{\frac{1}{2} \cdot 2} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3.3

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

$\frac{9 \sqrt{3} - 54 - 2 \cdot 3^{\frac{2}{2}} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{9 \sqrt{3} - 54 - 2 \cdot 3^{\frac{2}{2}} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{9 \sqrt{3} - 54 - 2 \cdot 3^{1} - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{9 \sqrt{3} - 54 - 2 \cdot 3 - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.4

Multiply $- 2$ by $3$ .

$\frac{9 \sqrt{3} - 54 - 6 - 2 \sqrt{3} \cdot - 6}{- 33}$

Step 4.1.5

Multiply $- 6$ by $- 2$ .

$\frac{9 \sqrt{3} - 54 - 6 + 12 \sqrt{3}}{- 33}$

Step 4.2

Add $9 \sqrt{3}$ and $12 \sqrt{3}$ .

$\frac{- 54 - 6 + 21 \sqrt{3}}{- 33}$

Step 4.3

Subtract $6$ from $- 54$ .

$\frac{- 60 + 21 \sqrt{3}}{- 33}$

Step 5

Cancel the common factor of $- 60 + 21 \sqrt{3}$ and $- 33$ .

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Step 5.1

Factor $3$ out of $- 60$ .

$\frac{3 (- 20) + 21 \sqrt{3}}{- 33}$

Step 5.2

Factor $3$ out of $21 \sqrt{3}$ .

$\frac{3 (- 20) + 3 (7 \sqrt{3})}{- 33}$

Step 5.3

Factor $3$ out of $3 (- 20) + 3 (7 \sqrt{3})$ .

$\frac{3 (- 20 + 7 \sqrt{3})}{- 33}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $3$ out of $- 33$ .

$\frac{3 (- 20 + 7 \sqrt{3})}{3 \cdot - 11}$

Step 5.4.2

Cancel the common factor.

$\frac{3 (- 20 + 7 \sqrt{3})}{3 \cdot - 11}$

Step 5.4.3

Rewrite the expression.

$\frac{- 20 + 7 \sqrt{3}}{- 11}$

Step 6

Move the negative in front of the fraction.

$- \frac{- 20 + 7 \sqrt{3}}{11}$

Step 7

Rewrite $- 20$ as $- 1 (20)$ .

$- \frac{- 1 (20) + 7 \sqrt{3}}{11}$

Step 8

Factor $- 1$ out of $7 \sqrt{3}$ .

$- \frac{- 1 (20) - (- 7 \sqrt{3})}{11}$

Step 9

Factor $- 1$ out of $- 1 (20) - (- 7 \sqrt{3})$ .

$- \frac{- 1 (20 - 7 \sqrt{3})}{11}$

Step 10

Simplify the expression.

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Step 10.1

Move the negative in front of the fraction.

$- - \frac{20 - 7 \sqrt{3}}{11}$

Step 10.2

Multiply $- 1$ by $- 1$ .

$1 \frac{20 - 7 \sqrt{3}}{11}$

Step 10.3

Multiply $\frac{20 - 7 \sqrt{3}}{11}$ by $1$ .

$\frac{20 - 7 \sqrt{3}}{11}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{20 - 7 \sqrt{3}}{11}$

Decimal Form:

$0.71596766 \dots$