Algebra Examples

$\frac{\sqrt{8} - \sqrt{3}}{\sqrt{2} + \sqrt{12}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{\sqrt{4 (2)} - \sqrt{3}}{\sqrt{2} + \sqrt{12}}$

Step 1.1.2

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

$\frac{\sqrt{2^{2} \cdot 2} - \sqrt{3}}{\sqrt{2} + \sqrt{12}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{12}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{4 (3)}}$

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Expand the denominator using the FOIL method.

$\frac{(2 \sqrt{2} - \sqrt{3}) (\sqrt{2} - 2 \sqrt{3})}{{\sqrt{2}}^{2} - 2 \sqrt{6} + 2 \sqrt{6} - 4 {\sqrt{3}}^{2}}$

Step 4.3

Simplify.

$\frac{(2 \sqrt{2} - \sqrt{3}) (\sqrt{2} - 2 \sqrt{3})}{- 10}$

Step 5

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

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Step 5.1

Apply the distributive property.

$\frac{2 \sqrt{2} (\sqrt{2} - 2 \sqrt{3}) - \sqrt{3} (\sqrt{2} - 2 \sqrt{3})}{- 10}$

Step 5.2

Apply the distributive property.

$\frac{2 \sqrt{2} \sqrt{2} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} (\sqrt{2} - 2 \sqrt{3})}{- 10}$

Step 5.3

Apply the distributive property.

$\frac{2 \sqrt{2} \sqrt{2} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

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 $2 \sqrt{2} \sqrt{2}$ .

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Step 6.1.1.1

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 ({\sqrt{2}}^{1} \sqrt{2}) + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.1.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.1.3

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

$\frac{2 {\sqrt{2}}^{1 + 1} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.1.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{2}}^{2} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2

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

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Step 6.1.2.1

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

$\frac{2 {(2^{\frac{1}{2}})}^{2} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2.2

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

$\frac{2 \cdot 2^{\frac{1}{2} \cdot 2} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2.3

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

$\frac{2 \cdot 2^{\frac{2}{2}} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

$\frac{2 \cdot 2^{\frac{2}{2}} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2.4.2

Rewrite the expression.

$\frac{2 \cdot 2^{1} + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.2.5

Evaluate the exponent.

$\frac{2 \cdot 2 + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.3

Multiply $2$ by $2$ .

$\frac{4 + 2 \sqrt{2} (- 2 \sqrt{3}) - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.4

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

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Step 6.1.4.1

Multiply $- 2$ by $2$ .

$\frac{4 - 4 \sqrt{2} \sqrt{3} - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.4.2

Combine using the product rule for radicals.

$\frac{4 - 4 \sqrt{3 \cdot 2} - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.4.3

Multiply $3$ by $2$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{3} \sqrt{2} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.5

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

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Step 6.1.5.1

Combine using the product rule for radicals.

$\frac{4 - 4 \sqrt{6} - \sqrt{2 \cdot 3} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.5.2

Multiply $2$ by $3$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{6} - \sqrt{3} (- 2 \sqrt{3})}{- 10}$

Step 6.1.6

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

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Step 6.1.6.1

Multiply $- 2$ by $- 1$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \sqrt{3} \sqrt{3}}{- 10}$

Step 6.1.6.2

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

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 ({\sqrt{3}}^{1} \sqrt{3})}{- 10}$

Step 6.1.6.3

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

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 10}$

Step 6.1.6.4

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

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 {\sqrt{3}}^{1 + 1}}{- 10}$

Step 6.1.6.5

Add $1$ and $1$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 {\sqrt{3}}^{2}}{- 10}$

Step 6.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 {(3^{\frac{1}{2}})}^{2}}{- 10}$

Step 6.1.7.2

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

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \cdot 3^{\frac{1}{2} \cdot 2}}{- 10}$

Step 6.1.7.3

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

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \cdot 3^{\frac{2}{2}}}{- 10}$

Step 6.1.7.4

Cancel the common factor of $2$ .

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Step 6.1.7.4.1

Cancel the common factor.

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \cdot 3^{\frac{2}{2}}}{- 10}$

Step 6.1.7.4.2

Rewrite the expression.

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \cdot 3^{1}}{- 10}$

Step 6.1.7.5

Evaluate the exponent.

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 2 \cdot 3}{- 10}$

Step 6.1.8

Multiply $2$ by $3$ .

$\frac{4 - 4 \sqrt{6} - \sqrt{6} + 6}{- 10}$

Step 6.2

Add $4$ and $6$ .

$\frac{10 - 4 \sqrt{6} - \sqrt{6}}{- 10}$

Step 6.3

Subtract $\sqrt{6}$ from $- 4 \sqrt{6}$ .

$\frac{10 - 5 \sqrt{6}}{- 10}$

Step 7

Cancel the common factor of $10 - 5 \sqrt{6}$ and $- 10$ .

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Step 7.1

Factor $5$ out of $10$ .

$\frac{5 (2) - 5 \sqrt{6}}{- 10}$

Step 7.2

Factor $5$ out of $- 5 \sqrt{6}$ .

$\frac{5 (2) + 5 (- \sqrt{6})}{- 10}$

Step 7.3

Factor $5$ out of $5 (2) + 5 (- \sqrt{6})$ .

$\frac{5 (2 - \sqrt{6})}{- 10}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $5$ out of $- 10$ .

$\frac{5 (2 - \sqrt{6})}{5 \cdot - 2}$

Step 7.4.2

Cancel the common factor.

$\frac{5 (2 - \sqrt{6})}{5 \cdot - 2}$

Step 7.4.3

Rewrite the expression.

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

Step 8

Move the negative in front of the fraction.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.22474487 \dots$