Simplify ( square root of 12- square root of 8)/( square root of 8- square root of 3)

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Algebra

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

Simplify the numerator.

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

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Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

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

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Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Simplify the denominator.

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

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Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

$\frac{(2 \sqrt{3} - 2 \sqrt{2}) (2 \sqrt{2} + \sqrt{3})}{5}$

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

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Apply the distributive property.

$\frac{2 \sqrt{3} (2 \sqrt{2} + \sqrt{3}) - 2 \sqrt{2} (2 \sqrt{2} + \sqrt{3})}{5}$

Apply the distributive property.

$\frac{2 \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{3} \sqrt{3} - 2 \sqrt{2} (2 \sqrt{2} + \sqrt{3})}{5}$

Apply the distributive property.

$\frac{2 \sqrt{3} (2 \sqrt{2}) + 2 \sqrt{3} \sqrt{3} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Simplify and combine like terms.

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Simplify each term.

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

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Multiply $2$ by $2$ .

$\frac{4 \sqrt{3} \sqrt{2} + 2 \sqrt{3} \sqrt{3} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Combine using the product rule for radicals.

$\frac{4 \sqrt{2 \cdot 3} + 2 \sqrt{3} \sqrt{3} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Multiply $2$ by $3$ .

$\frac{4 \sqrt{6} + 2 \sqrt{3} \sqrt{3} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

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

$\frac{4 \sqrt{6} + 2 ({\sqrt{3}}^{1} \sqrt{3}) - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 2 {\sqrt{3}}^{1 + 1} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Add $1$ and $1$ .

$\frac{4 \sqrt{6} + 2 {\sqrt{3}}^{2} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

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

$\frac{4 \sqrt{6} + 2 {(3^{\frac{1}{2}})}^{2} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 2 \cdot 3^{\frac{1}{2} \cdot 2} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 2 \cdot 3^{\frac{2}{2}} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{4 \sqrt{6} + 2 \cdot 3^{\frac{2}{2}} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Rewrite the expression.

$\frac{4 \sqrt{6} + 2 \cdot 3^{1} - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Evaluate the exponent.

$\frac{4 \sqrt{6} + 2 \cdot 3 - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

Multiply $2$ by $3$ .

$\frac{4 \sqrt{6} + 6 - 2 \sqrt{2} (2 \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

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Multiply $2$ by $- 2$ .

$\frac{4 \sqrt{6} + 6 - 4 \sqrt{2} \sqrt{2} - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 6 - 4 ({\sqrt{2}}^{1} \sqrt{2}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 6 - 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 6 - 4 {\sqrt{2}}^{1 + 1} - 2 \sqrt{2} \sqrt{3}}{5}$

Add $1$ and $1$ .

$\frac{4 \sqrt{6} + 6 - 4 {\sqrt{2}}^{2} - 2 \sqrt{2} \sqrt{3}}{5}$

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

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

$\frac{4 \sqrt{6} + 6 - 4 {(2^{\frac{1}{2}})}^{2} - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 6 - 4 \cdot 2^{\frac{1}{2} \cdot 2} - 2 \sqrt{2} \sqrt{3}}{5}$

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

$\frac{4 \sqrt{6} + 6 - 4 \cdot 2^{\frac{2}{2}} - 2 \sqrt{2} \sqrt{3}}{5}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{4 \sqrt{6} + 6 - 4 \cdot 2^{\frac{2}{2}} - 2 \sqrt{2} \sqrt{3}}{5}$

Rewrite the expression.

$\frac{4 \sqrt{6} + 6 - 4 \cdot 2^{1} - 2 \sqrt{2} \sqrt{3}}{5}$

Evaluate the exponent.

$\frac{4 \sqrt{6} + 6 - 4 \cdot 2 - 2 \sqrt{2} \sqrt{3}}{5}$

Multiply $- 4$ by $2$ .

$\frac{4 \sqrt{6} + 6 - 8 - 2 \sqrt{2} \sqrt{3}}{5}$

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

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Combine using the product rule for radicals.

$\frac{4 \sqrt{6} + 6 - 8 - 2 \sqrt{3 \cdot 2}}{5}$

Multiply $3$ by $2$ .

$\frac{4 \sqrt{6} + 6 - 8 - 2 \sqrt{6}}{5}$

Subtract $2 \sqrt{6}$ from $4 \sqrt{6}$ .

$\frac{6 - 8 + 2 \sqrt{6}}{5}$

Subtract $8$ from $6$ .

$\frac{- 2 + 2 \sqrt{6}}{5}$

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (2) + 2 \sqrt{6}}{5}$

Factor $- 1$ out of $2 \sqrt{6}$ .

$\frac{- 1 (2) - (- 2 \sqrt{6})}{5}$

Factor $- 1$ out of $- 1 (2) - (- 2 \sqrt{6})$ .

$\frac{- 1 (2 - 2 \sqrt{6})}{5}$

Move the negative in front of the fraction.

$- \frac{2 - 2 \sqrt{6}}{5}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 - 2 \sqrt{6}}{5}$

Decimal Form:

$0.57979589 \dots$

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