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

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Expand $(3 + \sqrt{3}) (\sqrt{3} + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $1$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

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

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

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

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

Add $3$ and $3$ .

$\frac{6 + 4 \sqrt{3}}{2}$

Step 5

Cancel the common factor of $6 + 4 \sqrt{3}$ and $2$ .

Factor $2$ out of $6$ .

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

Factor $2$ out of $4 \sqrt{3}$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$3 + 2 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$3 + 2 \sqrt{3}$

Decimal Form:

$6.46410161 \dots$

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

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