Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

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

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

Step 4.1.1.4

Add $1$ and $1$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.1

Cancel the common factor.

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

Step 4.1.2.4.2

Rewrite the expression.

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

Step 4.1.2.5

Evaluate the exponent.

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

Step 4.1.3

Multiply $3$ by $3$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Combine using the product rule for radicals.

$9 + 3 \sqrt{2 \cdot 3} - 2 \sqrt{2} \sqrt{3} - 2 \sqrt{2} \sqrt{2}$

Step 4.1.4.2

Multiply $2$ by $3$ .

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

Step 4.1.5

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

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Step 4.1.5.1

Combine using the product rule for radicals.

$9 + 3 \sqrt{6} - 2 \sqrt{3 \cdot 2} - 2 \sqrt{2} \sqrt{2}$

Step 4.1.5.2

Multiply $3$ by $2$ .

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

Step 4.1.6

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

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Step 4.1.6.1

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 ({\sqrt{2}}^{1} \sqrt{2})$

Step 4.1.6.2

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 4.1.6.3

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 {\sqrt{2}}^{1 + 1}$

Step 4.1.6.4

Add $1$ and $1$ .

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 {\sqrt{2}}^{2}$

Step 4.1.7

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

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Step 4.1.7.1

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 {(2^{\frac{1}{2}})}^{2}$

Step 4.1.7.2

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 4.1.7.3

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

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 \cdot 2^{\frac{2}{2}}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 \cdot 2^{\frac{2}{2}}$

Step 4.1.7.4.2

Rewrite the expression.

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 \cdot 2^{1}$

Step 4.1.7.5

Evaluate the exponent.

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 2 \cdot 2$

Step 4.1.8

Multiply $- 2$ by $2$ .

$9 + 3 \sqrt{6} - 2 \sqrt{6} - 4$

Step 4.2

Subtract $4$ from $9$ .

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

Step 4.3

Subtract $2 \sqrt{6}$ from $3 \sqrt{6}$ .

$5 + \sqrt{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$5 + \sqrt{6}$

Decimal Form:

$7.44948974 \dots$