Algebra Examples

$\frac{\sqrt{27} - \sqrt{6}}{\sqrt{18} - 3}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1.1

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)} - \sqrt{6}}{\sqrt{18} - 3}$

Step 1.1.2

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

$\frac{\sqrt{3^{2} \cdot 3} - \sqrt{6}}{\sqrt{18} - 3}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 \sqrt{3} - \sqrt{6}}{\sqrt{18} - 3}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

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

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

Multiply $\frac{3 \sqrt{3} - \sqrt{6}}{3 \sqrt{2} - 3}$ by $\frac{3 \sqrt{2} + 3}{3 \sqrt{2} + 3}$ .

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

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{3 \sqrt{3} - \sqrt{6}}{3 \sqrt{2} - 3}$ by $\frac{3 \sqrt{2} + 3}{3 \sqrt{2} + 3}$ .

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

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

Step 4.4

Cancel the common factor of $3 \sqrt{2} + 3$ and $9$ .

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $3$ out of $9$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 5

Expand $(3 \sqrt{3} - \sqrt{6}) (\sqrt{2} + 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $3 \sqrt{3} \sqrt{2}$ .

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Step 6.1.1.1

Combine using the product rule for radicals.

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

Step 6.1.1.2

Multiply $2$ by $3$ .

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

Step 6.1.2

Multiply $3$ by $1$ .

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

Step 6.1.3

Multiply $- \sqrt{6} \sqrt{2}$ .

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Step 6.1.3.1

Combine using the product rule for radicals.

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

Step 6.1.3.2

Multiply $2$ by $6$ .

$\frac{3 \sqrt{6} + 3 \sqrt{3} - \sqrt{12} - \sqrt{6} \cdot 1}{3}$

Step 6.1.4

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

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Step 6.1.4.1

Factor $4$ out of $12$ .

$\frac{3 \sqrt{6} + 3 \sqrt{3} - \sqrt{4 (3)} - \sqrt{6} \cdot 1}{3}$

Step 6.1.4.2

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

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

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.1.6

Multiply $2$ by $- 1$ .

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

Step 6.1.7

Multiply $- 1$ by $1$ .

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

Step 6.2

Subtract $\sqrt{6}$ from $3 \sqrt{6}$ .

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

Step 6.3

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.21034343 \dots$