Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $6$ by $2$ .

$\frac{12 \sqrt{3} + 2 \sqrt{3} \sqrt{3} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

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 $1$ .

$\frac{12 \sqrt{3} + 2 ({\sqrt{3}}^{1} \sqrt{3}) - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.2.2

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

$\frac{12 \sqrt{3} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.2.3

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

$\frac{12 \sqrt{3} + 2 {\sqrt{3}}^{1 + 1} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{12 \sqrt{3} + 2 {\sqrt{3}}^{2} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

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 $3^{\frac{1}{2}}$ .

$\frac{12 \sqrt{3} + 2 {(3^{\frac{1}{2}})}^{2} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.3.2

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

$\frac{12 \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.3.3

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

$\frac{12 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

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.

$\frac{12 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{12 \sqrt{3} + 2 \cdot 3^{1} - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{12 \sqrt{3} + 2 \cdot 3 - 3 \cdot 6 - 3 \sqrt{3}}{33}$

Step 4.1.4

Multiply $2$ by $3$ .

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

Step 4.1.5

Multiply $- 3$ by $6$ .

$\frac{12 \sqrt{3} + 6 - 18 - 3 \sqrt{3}}{33}$

Step 4.2

Subtract $3 \sqrt{3}$ from $12 \sqrt{3}$ .

$\frac{6 - 18 + 9 \sqrt{3}}{33}$

Step 4.3

Subtract $18$ from $6$ .

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

Step 5

Cancel the common factor of $- 12 + 9 \sqrt{3}$ and $33$ .

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Step 5.1

Factor $3$ out of $- 12$ .

$\frac{3 (- 4) + 9 \sqrt{3}}{33}$

Step 5.2

Factor $3$ out of $9 \sqrt{3}$ .

$\frac{3 (- 4) + 3 (3 \sqrt{3})}{33}$

Step 5.3

Factor $3$ out of $3 (- 4) + 3 (3 \sqrt{3})$ .

$\frac{3 (- 4 + 3 \sqrt{3})}{33}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $3$ out of $33$ .

$\frac{3 (- 4 + 3 \sqrt{3})}{3 \cdot 11}$

Step 5.4.2

Cancel the common factor.

$\frac{3 (- 4 + 3 \sqrt{3})}{3 \cdot 11}$

Step 5.4.3

Rewrite the expression.

$\frac{- 4 + 3 \sqrt{3}}{11}$

Step 6

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- 1 (4) + 3 \sqrt{3}}{11}$

Step 7

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

$\frac{- 1 (4) - (- 3 \sqrt{3})}{11}$

Step 8

Factor $- 1$ out of $- 1 (4) - (- 3 \sqrt{3})$ .

$\frac{- 1 (4 - 3 \sqrt{3})}{11}$

Step 9

Move the negative in front of the fraction.

$- \frac{4 - 3 \sqrt{3}}{11}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 - 3 \sqrt{3}}{11}$

Decimal Form:

$0.10874112 \dots$