Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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

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Step 2.1.1.1

Multiply $3$ by $2$ .

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

Step 2.1.1.2

Combine using the product rule for radicals.

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

Step 2.1.1.3

Multiply $2$ by $3$ .

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

Step 2.1.2

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

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Step 2.1.2.1

Multiply $- 1$ by $2$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

Add $1$ and $1$ .

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Cancel the common factor.

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

Step 2.1.3.4.2

Rewrite the expression.

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

Step 2.1.3.5

Evaluate the exponent.

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

Step 2.1.4

Multiply $- 2$ by $3$ .

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

Step 2.1.5

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

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Add $1$ and $1$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

Step 2.1.7

Multiply $3$ by $2$ .

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

Step 2.1.8

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

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Step 2.1.8.1

Combine using the product rule for radicals.

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

Step 2.1.8.2

Multiply $2$ by $3$ .

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

Step 2.2

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

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

Step 2.3

Add $- 6$ and $6$ .

$0 + 5 \sqrt{6}$

Step 2.4

Add $0$ and $5 \sqrt{6}$ .

$5 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{6}$

Decimal Form:

$12.24744871 \dots$