Algebra Examples

$\frac{\sqrt{3} - 2}{4 + \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{3} - 2}{4 + \sqrt{3}}$ by $\frac{4 - \sqrt{3}}{4 - \sqrt{3}}$ .

$\frac{\sqrt{3} - 2}{4 + \sqrt{3}} \cdot \frac{4 - \sqrt{3}}{4 - \sqrt{3}}$

Step 2

Multiply $\frac{\sqrt{3} - 2}{4 + \sqrt{3}}$ by $\frac{4 - \sqrt{3}}{4 - \sqrt{3}}$ .

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

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - 2) (4 - \sqrt{3})}{16 - 4 \sqrt{3} + \sqrt{3} \cdot 4 - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{(\sqrt{3} - 2) (4 - \sqrt{3})}{13}$

Step 5

Expand $(\sqrt{3} - 2) (4 - \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{3} (4 - \sqrt{3}) - 2 (4 - \sqrt{3})}{13}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{3} \cdot 4 + \sqrt{3} (- \sqrt{3}) - 2 (4 - \sqrt{3})}{13}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{3} \cdot 4 + \sqrt{3} (- \sqrt{3}) - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Move $4$ to the left of $\sqrt{3}$ .

$\frac{4 \cdot \sqrt{3} + \sqrt{3} (- \sqrt{3}) - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.2

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

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Step 6.1.2.1

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

$\frac{4 \sqrt{3} - ({\sqrt{3}}^{1} \sqrt{3}) - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.2.2

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

$\frac{4 \sqrt{3} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.2.3

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

$\frac{4 \sqrt{3} - {\sqrt{3}}^{1 + 1} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.2.4

Add $1$ and $1$ .

$\frac{4 \sqrt{3} - {\sqrt{3}}^{2} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3

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

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Step 6.1.3.1

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

$\frac{4 \sqrt{3} - {(3^{\frac{1}{2}})}^{2} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3.2

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

$\frac{4 \sqrt{3} - 3^{\frac{1}{2} \cdot 2} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3.3

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

$\frac{4 \sqrt{3} - 3^{\frac{2}{2}} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3.4

Cancel the common factor of $2$ .

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Step 6.1.3.4.1

Cancel the common factor.

$\frac{4 \sqrt{3} - 3^{\frac{2}{2}} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3.4.2

Rewrite the expression.

$\frac{4 \sqrt{3} - 3^{1} - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.3.5

Evaluate the exponent.

$\frac{4 \sqrt{3} - 1 \cdot 3 - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.4

Multiply $- 1$ by $3$ .

$\frac{4 \sqrt{3} - 3 - 2 \cdot 4 - 2 (- \sqrt{3})}{13}$

Step 6.1.5

Multiply $- 2$ by $4$ .

$\frac{4 \sqrt{3} - 3 - 8 - 2 (- \sqrt{3})}{13}$

Step 6.1.6

Multiply $- 1$ by $- 2$ .

$\frac{4 \sqrt{3} - 3 - 8 + 2 \sqrt{3}}{13}$

Step 6.2

Add $4 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{- 3 - 8 + 6 \sqrt{3}}{13}$

Step 6.3

Subtract $8$ from $- 3$ .

$\frac{- 11 + 6 \sqrt{3}}{13}$

Step 7

Rewrite $- 11$ as $- 1 (11)$ .

$\frac{- 1 (11) + 6 \sqrt{3}}{13}$

Step 8

Factor $- 1$ out of $6 \sqrt{3}$ .

$\frac{- 1 (11) - (- 6 \sqrt{3})}{13}$

Step 9

Factor $- 1$ out of $- 1 (11) - (- 6 \sqrt{3})$ .

$\frac{- 1 (11 - 6 \sqrt{3})}{13}$

Step 10

Move the negative in front of the fraction.

$- \frac{11 - 6 \sqrt{3}}{13}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{11 - 6 \sqrt{3}}{13}$

Decimal Form:

$- 0.04674578 \dots$