Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Evaluate the exponent.

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

Step 2.1.3

Multiply $- 1$ by $3$ .

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

Step 2.1.4

Combine using the product rule for radicals.

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

Step 2.1.5

Multiply $3$ by $2$ .

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

Step 2.1.6

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

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Step 2.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.6.2

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

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

Step 2.1.6.3

Combine using the product rule for radicals.

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

Step 2.1.6.4

Multiply $2$ by $3$ .

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

Step 2.1.7

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

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Step 2.1.7.1

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

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

Step 2.1.7.2

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

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

Step 2.1.7.3

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

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

Step 2.1.7.4

Add $1$ and $1$ .

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

Step 2.1.8

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

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Step 2.1.8.1

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 2.1.8.2

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 2.1.8.3

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

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

Step 2.1.8.4

Cancel the common factor of $2$ .

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Step 2.1.8.4.1

Cancel the common factor.

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

Step 2.1.8.4.2

Rewrite the expression.

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

Step 2.1.8.5

Evaluate the exponent.

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

Step 2.1.9

Multiply $- 1$ by $2$ .

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

Step 2.2

Subtract $2$ from $- 3$ .

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

Step 2.3

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

$- 5 + 2 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 5 + 2 \sqrt{6}$

Decimal Form:

$- 0.10102051 \dots$