Simplify ( square root of 3+1)/(1- square root of 3)*( square root of 3+1)/( square root of 3+1)

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

Cancel the common factor of $\sqrt{3} + 1$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify terms.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Move the negative in front of the fraction.

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

Apply the distributive property.

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

Combine $\sqrt{3}$ and $\frac{1 + \sqrt{3}}{2}$ .

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

Simplify the expression.

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Multiply $- 1$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

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

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

Subtract $1$ from $- 3$ .

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

Cancel the common factor of $- 4 - 2 \sqrt{3}$ and $2$ .

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Factor $2$ out of $- 4$ .

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

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

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

Factor $2$ out of $2 (- 2) + 2 (- \sqrt{3})$ .

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

Cancel the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2 - \sqrt{3}$ by $1$ .

$- 2 - \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$

Simplify ( square root of 3+1)/(1- square root of 3)*( square root of 3+1)/( square root of 3+1)

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