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

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

$\frac{(3 - \sqrt{3}) (3 - \sqrt{3})}{9 - 3 \sqrt{3} + \sqrt{3} \cdot 3 - {\sqrt{3}}^{2}}$

Simplify.

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

Step 3

Simplify the numerator.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Rewrite ${(3 - \sqrt{3})}^{2}$ as $(3 - \sqrt{3}) (3 - \sqrt{3})$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$\frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Multiply $- 1$ by $3$ .

$\frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Multiply $3$ by $- 1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

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

Multiply $- 1$ by $- 1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Add $1$ and $1$ .

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{2}}{6}$

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

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}$

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

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{2}{2}}}{6}$

Rewrite the expression.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Evaluate the exponent.

$\frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3}{6}$

Add $9$ and $3$ .

$\frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}$

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

$\frac{12 - 6 \sqrt{3}}{6}$

Step 7

Cancel the common factor of $12 - 6 \sqrt{3}$ and $6$ .

Factor $6$ out of $12$ .

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

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

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

Factor $6$ out of $6 (2) + 6 (- \sqrt{3})$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$2 - \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{3}$

Decimal Form:

$0.26794919 \dots$

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

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