Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - \sqrt{6}) (\sqrt{3} - \sqrt{6})}{{\sqrt{3}}^{2} - \sqrt{18} + \sqrt{18} - {\sqrt{6}}^{2}}$

Step 4

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $\sqrt{3} - \sqrt{6}$ to the power of $1$ .

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

Step 5.2

Raise $\sqrt{3} - \sqrt{6}$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Rewrite ${(\sqrt{3} - \sqrt{6})}^{2}$ as $(\sqrt{3} - \sqrt{6}) (\sqrt{3} - \sqrt{6})$ .

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Combine using the product rule for radicals.

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

Step 8.1.2

Multiply $3$ by $3$ .

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

Step 8.1.3

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

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

Step 8.1.4

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

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

Step 8.1.5

Multiply $\sqrt{3} (- \sqrt{6})$ .

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Step 8.1.5.1

Combine using the product rule for radicals.

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

Step 8.1.5.2

Multiply $3$ by $6$ .

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

Step 8.1.6

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

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Step 8.1.6.1

Factor $9$ out of $18$ .

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

Step 8.1.6.2

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

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

Step 8.1.7

Pull terms out from under the radical.

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

Step 8.1.8

Multiply $3$ by $- 1$ .

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

Step 8.1.9

Multiply $- \sqrt{6} \sqrt{3}$ .

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Step 8.1.9.1

Combine using the product rule for radicals.

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

Step 8.1.9.2

Multiply $3$ by $6$ .

$\frac{3 - 3 \sqrt{2} - \sqrt{18} - \sqrt{6} (- \sqrt{6})}{- 3}$

Step 8.1.10

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

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Step 8.1.10.1

Factor $9$ out of $18$ .

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

Step 8.1.10.2

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

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

Step 8.1.11

Pull terms out from under the radical.

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

Step 8.1.12

Multiply $3$ by $- 1$ .

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

Step 8.1.13

Multiply $- \sqrt{6} (- \sqrt{6})$ .

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Step 8.1.13.1

Multiply $- 1$ by $- 1$ .

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

Step 8.1.13.2

Multiply $\sqrt{6}$ by $1$ .

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

Step 8.1.13.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 8.1.13.4

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 8.1.13.5

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

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

Step 8.1.13.6

Add $1$ and $1$ .

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

Step 8.1.14

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 8.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 8.1.14.2

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

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

Step 8.1.14.3

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

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

Step 8.1.14.4

Cancel the common factor of $2$ .

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Step 8.1.14.4.1

Cancel the common factor.

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

Step 8.1.14.4.2

Rewrite the expression.

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

Step 8.1.14.5

Evaluate the exponent.

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

Step 8.2

Add $3$ and $6$ .

$\frac{9 - 3 \sqrt{2} - 3 \sqrt{2}}{- 3}$

Step 8.3

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

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

Step 9

Cancel the common factor of $9 - 6 \sqrt{2}$ and $- 3$ .

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Step 9.1

Factor $3$ out of $9$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Move the negative one from the denominator of $\frac{3 - 2 \sqrt{2}}{- 1}$ .

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

Step 10

Rewrite $- 1 \cdot (3 - 2 \sqrt{2})$ as $- (3 - 2 \sqrt{2})$ .

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

Step 11

Apply the distributive property.

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

Step 12

Multiply $- 1$ by $3$ .

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

Step 13

Multiply $- 2$ by $- 1$ .

$- 3 + 2 \sqrt{2}$