Basic Math Examples

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

Step 1

Expand $(\sqrt{3} - \sqrt{6}) (\sqrt{2} - \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Combine using the product rule for radicals.

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

Step 2.2

Multiply $3$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Add $1$ and $1$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

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

Step 2.4.4.2

Rewrite the expression.

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

Step 2.4.5

Evaluate the exponent.

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

Step 2.5

Multiply $- 1$ by $3$ .

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

Step 2.6

Multiply $- \sqrt{6} \sqrt{2}$ .

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Step 2.6.1

Combine using the product rule for radicals.

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

Step 2.6.2

Multiply $2$ by $6$ .

$\sqrt{6} - 3 - \sqrt{12} - \sqrt{6} (- \sqrt{3})$

Step 2.7

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

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Step 2.7.1

Factor $4$ out of $12$ .

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

Step 2.7.2

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

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

Step 2.8

Pull terms out from under the radical.

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

Step 2.9

Multiply $2$ by $- 1$ .

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

Step 2.10

Multiply $- \sqrt{6} (- \sqrt{3})$ .

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

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

Step 2.10.2

Multiply $\sqrt{6}$ by $1$ .

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

Step 2.10.3

Combine using the product rule for radicals.

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

Step 2.10.4

Multiply $6$ by $3$ .

$\sqrt{6} - 3 - 2 \sqrt{3} + \sqrt{18}$

Step 2.11

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.11.1

Factor $9$ out of $18$ .

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

Step 2.11.2

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

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

Step 2.12

Pull terms out from under the radical.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.22802881 \dots$