Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Divide $(\sqrt{3} - \sqrt{2}) (\sqrt{3} - \sqrt{2})$ by $1$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

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

Step 4.1.2

Multiply $3$ by $3$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

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Step 4.1.5.1

Combine using the product rule for radicals.

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

Step 4.1.5.2

Multiply $3$ by $2$ .

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

Step 4.1.6

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

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Step 4.1.6.1

Combine using the product rule for radicals.

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

Step 4.1.6.2

Multiply $3$ by $2$ .

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

Step 4.1.7

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

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.7.2

Multiply $\sqrt{2}$ by $1$ .

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

Step 4.1.7.3

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

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

Step 4.1.7.4

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

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

Step 4.1.7.5

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

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

Step 4.1.7.6

Add $1$ and $1$ .

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

Step 4.1.8

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

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Step 4.1.8.1

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

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

Step 4.1.8.2

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

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

Step 4.1.8.3

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

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

Step 4.1.8.4

Cancel the common factor of $2$ .

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Step 4.1.8.4.1

Cancel the common factor.

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

Step 4.1.8.4.2

Rewrite the expression.

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

Step 4.1.8.5

Evaluate the exponent.

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

Step 4.2

Add $3$ and $2$ .

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

Step 4.3

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

$5 - 2 \sqrt{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$5 - 2 \sqrt{6}$

Decimal Form:

$0.10102051 \dots$