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.

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

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 $3$ .

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

Move $3$ to the left of $\sqrt{3}$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Add $9$ and $3$ .

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

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

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

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:

$3.73205080 \dots$

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

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