Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1

Factor $4$ out of $12$ .

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

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Expand the denominator using the FOIL method.

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

Step 3.3

Simplify.

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

Step 3.4

Cancel the common factor of $2$ and $- 6$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $- 6$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Combine using the product rule for radicals.

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

Step 3.7

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{\sqrt{9} - 3 \cdot \sqrt{3}}{- 3}$

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Factor $3$ out of $3$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Move the negative one from the denominator of $\frac{1 - \sqrt{3}}{- 1}$ .

$- 1 \cdot (1 - \sqrt{3})$

Step 5.2

Rewrite $- 1 \cdot (1 - \sqrt{3})$ as $- (1 - \sqrt{3})$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $- 1$ by $1$ .

$- 1 - - \sqrt{3}$

Step 6

Multiply $- - \sqrt{3}$ .

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Step 6.1

Multiply $- 1$ by $- 1$ .

$- 1 + 1 \sqrt{3}$

Step 6.2

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

$- 1 + \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 1 + \sqrt{3}$

Decimal Form:

$0.73205080 \dots$