Algebra Examples

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

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 $8$ .

$\frac{\sqrt{3} - \sqrt{4 (2)}}{\sqrt{8} + \sqrt{12}}$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $8$ .

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

Step 2.1.2

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

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

Step 2.2

Pull terms out from under the radical.

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

Step 2.3

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

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Step 2.3.1

Factor $4$ out of $12$ .

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

Step 2.3.2

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

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

Step 2.4

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

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

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $- 4$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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

Combine using the product rule for radicals.

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

Step 6.1.2

Multiply $3$ by $2$ .

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

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

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

Step 6.1.3.3

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

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

Step 6.1.3.4

Add $1$ and $1$ .

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

Step 6.1.4

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

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Step 6.1.4.1

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

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

Step 6.1.4.2

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

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

Step 6.1.4.3

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

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

Step 6.1.4.4

Cancel the common factor of $2$ .

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Step 6.1.4.4.1

Cancel the common factor.

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

Step 6.1.4.4.2

Rewrite the expression.

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

Step 6.1.4.5

Evaluate the exponent.

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

Step 6.1.5

Multiply $- 1$ by $3$ .

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

Step 6.1.6

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

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Step 6.1.6.1

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

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

Step 6.1.6.2

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

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

Step 6.1.6.3

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

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

Step 6.1.6.4

Add $1$ and $1$ .

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

Step 6.1.7

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

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Step 6.1.7.1

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

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

Step 6.1.7.2

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

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

Step 6.1.7.3

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

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

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.

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

Step 6.1.7.4.2

Rewrite the expression.

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

Step 6.1.7.5

Evaluate the exponent.

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

Step 6.1.8

Multiply $- 2$ by $2$ .

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

Step 6.1.9

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

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Step 6.1.9.1

Multiply $- 1$ by $- 2$ .

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

Step 6.1.9.2

Combine using the product rule for radicals.

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

Step 6.1.9.3

Multiply $3$ by $2$ .

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

Step 6.2

Add $\sqrt{6}$ and $2 \sqrt{6}$ .

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

Step 6.3

Subtract $4$ from $- 3$ .

$\frac{- 7 + 3 \sqrt{6}}{- 2}$

Step 7

Move the negative in front of the fraction.

$- \frac{- 7 + 3 \sqrt{6}}{2}$

Step 8

Rewrite $- 7$ as $- 1 (7)$ .

$- \frac{- 1 (7) + 3 \sqrt{6}}{2}$

Step 9

Factor $- 1$ out of $3 \sqrt{6}$ .

$- \frac{- 1 (7) - (- 3 \sqrt{6})}{2}$

Step 10

Factor $- 1$ out of $- 1 (7) - (- 3 \sqrt{6})$ .

$- \frac{- 1 (7 - 3 \sqrt{6})}{2}$

Step 11

Simplify the expression.

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Step 11.1

Move the negative in front of the fraction.

$- - \frac{7 - 3 \sqrt{6}}{2}$

Step 11.2

Multiply $- 1$ by $- 1$ .

$1 \frac{7 - 3 \sqrt{6}}{2}$

Step 11.3

Multiply $\frac{7 - 3 \sqrt{6}}{2}$ by $1$ .

$\frac{7 - 3 \sqrt{6}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{7 - 3 \sqrt{6}}{2}$

Decimal Form:

$- 0.17423461 \dots$