Evaluate (3- square root of 3)/(4-2 square root of 3)

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

Step 1

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

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

Step 2

Simplify terms.

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

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

Expand the denominator using the FOIL method.

$\frac{(3 - \sqrt{3}) (4 + 2 \sqrt{3})}{16 + 8 \sqrt{3} - 8 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Simplify.

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

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

Factor $2$ out of $(3 - \sqrt{3}) (4 + 2 \sqrt{3})$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $2$ .

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

Multiply $2$ by $- 1$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $6$ .

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.36602540 \dots$

Evaluate (3- square root of 3)/(4-2 square root of 3)

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