Basic Math Examples

$\frac{1 - \sqrt{3}}{\sqrt{3} - 2}$

Step 1

Multiply $\frac{1 - \sqrt{3}}{\sqrt{3} - 2}$ by $\frac{\sqrt{3} + 2}{\sqrt{3} + 2}$ .

$\frac{1 - \sqrt{3}}{\sqrt{3} - 2} \cdot \frac{\sqrt{3} + 2}{\sqrt{3} + 2}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 - \sqrt{3}}{\sqrt{3} - 2}$ by $\frac{\sqrt{3} + 2}{\sqrt{3} + 2}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{(1 - \sqrt{3}) (\sqrt{3} + 2)}{- 1}$ .

$- 1 \cdot ((1 - \sqrt{3}) (\sqrt{3} + 2))$

Step 2.4.2

Rewrite $- 1 \cdot ((1 - \sqrt{3}) (\sqrt{3} + 2))$ as $- ((1 - \sqrt{3}) (\sqrt{3} + 2))$ .

$- ((1 - \sqrt{3}) (\sqrt{3} + 2))$

Step 3

Expand $(1 - \sqrt{3}) (\sqrt{3} + 2)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$- (1 (\sqrt{3} + 2) - \sqrt{3} (\sqrt{3} + 2))$

Step 3.2

Apply the distributive property.

$- (1 \sqrt{3} + 1 \cdot 2 - \sqrt{3} (\sqrt{3} + 2))$

Step 3.3

Apply the distributive property.

$- (1 \sqrt{3} + 1 \cdot 2 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot 2)$

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

$- (\sqrt{3} + 1 \cdot 2 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot 2)$

Step 4.1.2

Multiply $2$ by $1$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

Add $1$ and $1$ .

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

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

Step 4.1.4.4.2

Rewrite the expression.

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

Step 4.1.4.5

Evaluate the exponent.

$- (\sqrt{3} + 2 - 1 \cdot 3 - \sqrt{3} \cdot 2)$

Step 4.1.5

Multiply $- 1$ by $3$ .

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

Step 4.1.6

Multiply $2$ by $- 1$ .

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

Step 4.2

Subtract $2 \sqrt{3}$ from $\sqrt{3}$ .

$- (2 - 3 - \sqrt{3})$

Step 4.3

Subtract $3$ from $2$ .

$- (- 1 - \sqrt{3})$

Step 5

Apply the distributive property.

$- - 1 - - \sqrt{3}$

Step 6

Multiply $- 1$ by $- 1$ .

$1 - - \sqrt{3}$

Step 7

Multiply $- - \sqrt{3}$ .

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Step 7.1

Multiply $- 1$ by $- 1$ .

$1 + 1 \sqrt{3}$

Step 7.2

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

$1 + \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$1 + \sqrt{3}$

Decimal Form:

$2.73205080 \dots$