Simplify (6- square root of 3)/( square root of 3+4)

$\frac{6 - \sqrt{3}}{\sqrt{3} + 4}$

Step 1

Multiply $\frac{6 - \sqrt{3}}{\sqrt{3} + 4}$ by $\frac{\sqrt{3} - 4}{\sqrt{3} - 4}$ .

$\frac{6 - \sqrt{3}}{\sqrt{3} + 4} \cdot \frac{\sqrt{3} - 4}{\sqrt{3} - 4}$

Step 2

Combine fractions.

Multiply $\frac{6 - \sqrt{3}}{\sqrt{3} + 4}$ by $\frac{\sqrt{3} - 4}{\sqrt{3} - 4}$ .

$\frac{(6 - \sqrt{3}) (\sqrt{3} - 4)}{(\sqrt{3} + 4) (\sqrt{3} - 4)}$

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{3}) (\sqrt{3} - 4)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 4 + 4 \sqrt{3} - 16}$

Simplify.

$\frac{(6 - \sqrt{3}) (\sqrt{3} - 4)}{- 13}$

Step 3

Expand $(6 - \sqrt{3}) (\sqrt{3} - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 (\sqrt{3} - 4) - \sqrt{3} (\sqrt{3} - 4)}{- 13}$

Apply the distributive property.

$\frac{6 \sqrt{3} + 6 \cdot - 4 - \sqrt{3} (\sqrt{3} - 4)}{- 13}$

Apply the distributive property.

$\frac{6 \sqrt{3} + 6 \cdot - 4 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot - 4}{- 13}$

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $- 4$ .

$\frac{6 \sqrt{3} - 24 - \sqrt{3} \sqrt{3} - \sqrt{3} \cdot - 4}{- 13}$

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

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

$\frac{6 \sqrt{3} - 24 - ({\sqrt{3}}^{1} \sqrt{3}) - \sqrt{3} \cdot - 4}{- 13}$

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

$\frac{6 \sqrt{3} - 24 - ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - \sqrt{3} \cdot - 4}{- 13}$

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

$\frac{6 \sqrt{3} - 24 - {\sqrt{3}}^{1 + 1} - \sqrt{3} \cdot - 4}{- 13}$

Add $1$ and $1$ .

$\frac{6 \sqrt{3} - 24 - {\sqrt{3}}^{2} - \sqrt{3} \cdot - 4}{- 13}$

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

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

$\frac{6 \sqrt{3} - 24 - {(3^{\frac{1}{2}})}^{2} - \sqrt{3} \cdot - 4}{- 13}$

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

$\frac{6 \sqrt{3} - 24 - 3^{\frac{1}{2} \cdot 2} - \sqrt{3} \cdot - 4}{- 13}$

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

$\frac{6 \sqrt{3} - 24 - 3^{\frac{2}{2}} - \sqrt{3} \cdot - 4}{- 13}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 \sqrt{3} - 24 - 3^{\frac{2}{2}} - \sqrt{3} \cdot - 4}{- 13}$

Rewrite the expression.

$\frac{6 \sqrt{3} - 24 - 3^{1} - \sqrt{3} \cdot - 4}{- 13}$

Evaluate the exponent.

$\frac{6 \sqrt{3} - 24 - 1 \cdot 3 - \sqrt{3} \cdot - 4}{- 13}$

Multiply $- 1$ by $3$ .

$\frac{6 \sqrt{3} - 24 - 3 - \sqrt{3} \cdot - 4}{- 13}$

Multiply $- 4$ by $- 1$ .

$\frac{6 \sqrt{3} - 24 - 3 + 4 \sqrt{3}}{- 13}$

Add $6 \sqrt{3}$ and $4 \sqrt{3}$ .

$\frac{- 24 - 3 + 10 \sqrt{3}}{- 13}$

Subtract $3$ from $- 24$ .

$\frac{- 27 + 10 \sqrt{3}}{- 13}$

Step 5

Move the negative in front of the fraction.

$- \frac{- 27 + 10 \sqrt{3}}{13}$

Step 6

Rewrite $- 27$ as $- 1 (27)$ .

$- \frac{- 1 (27) + 10 \sqrt{3}}{13}$

Step 7

Factor $- 1$ out of $10 \sqrt{3}$ .

$- \frac{- 1 (27) - (- 10 \sqrt{3})}{13}$

Step 8

Factor $- 1$ out of $- 1 (27) - (- 10 \sqrt{3})$ .

$- \frac{- 1 (27 - 10 \sqrt{3})}{13}$

Step 9

Simplify the expression.

Move the negative in front of the fraction.

$- - \frac{27 - 10 \sqrt{3}}{13}$

Multiply $- 1$ by $- 1$ .

$1 \frac{27 - 10 \sqrt{3}}{13}$

Multiply $\frac{27 - 10 \sqrt{3}}{13}$ by $1$ .

$\frac{27 - 10 \sqrt{3}}{13}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{27 - 10 \sqrt{3}}{13}$

Decimal Form:

$0.74457630 \dots$

Simplify (6- square root of 3)/( square root of 3+4)

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