Basic Math Examples

$\frac{2 \sqrt{3 \cdot 4} + 3 \sqrt{6}}{3 - \sqrt{6}}$

Step 1

Simplify the numerator.

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Step 1.1

Multiply $3$ by $4$ .

$\frac{2 \sqrt{12} + 3 \sqrt{6}}{3 - \sqrt{6}}$

Step 1.2

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

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Step 1.2.1

Factor $4$ out of $12$ .

$\frac{2 \sqrt{4 (3)} + 3 \sqrt{6}}{3 - \sqrt{6}}$

Step 1.2.2

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

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

Step 1.3

Pull terms out from under the radical.

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

Step 1.4

Multiply $2$ by $2$ .

$\frac{4 \sqrt{3} + 3 \sqrt{6}}{3 - \sqrt{6}}$

Step 2

Multiply $\frac{4 \sqrt{3} + 3 \sqrt{6}}{3 - \sqrt{6}}$ by $\frac{3 + \sqrt{6}}{3 + \sqrt{6}}$ .

$\frac{4 \sqrt{3} + 3 \sqrt{6}}{3 - \sqrt{6}} \cdot \frac{3 + \sqrt{6}}{3 + \sqrt{6}}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{4 \sqrt{3} + 3 \sqrt{6}}{3 - \sqrt{6}}$ by $\frac{3 + \sqrt{6}}{3 + \sqrt{6}}$ .

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(4 \sqrt{3} + 3 \sqrt{6}) (3 + \sqrt{6})}{9 + 3 \sqrt{6} - 3 \sqrt{6} - {\sqrt{6}}^{2}}$

Step 3.3

Simplify.

$\frac{(4 \sqrt{3} + 3 \sqrt{6}) (3 + \sqrt{6})}{3}$

Step 4

Expand $(4 \sqrt{3} + 3 \sqrt{6}) (3 + \sqrt{6})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$\frac{4 \sqrt{3} (3 + \sqrt{6}) + 3 \sqrt{6} (3 + \sqrt{6})}{3}$

Step 4.2

Apply the distributive property.

$\frac{4 \sqrt{3} \cdot 3 + 4 \sqrt{3} \sqrt{6} + 3 \sqrt{6} (3 + \sqrt{6})}{3}$

Step 4.3

Apply the distributive property.

$\frac{4 \sqrt{3} \cdot 3 + 4 \sqrt{3} \sqrt{6} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5

Simplify terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $3$ by $4$ .

$\frac{12 \sqrt{3} + 4 \sqrt{3} \sqrt{6} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.2

Multiply $4 \sqrt{3} \sqrt{6}$ .

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Step 5.1.2.1

Combine using the product rule for radicals.

$\frac{12 \sqrt{3} + 4 \sqrt{6 \cdot 3} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.2.2

Multiply $6$ by $3$ .

$\frac{12 \sqrt{3} + 4 \sqrt{18} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.3

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

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Step 5.1.3.1

Factor $9$ out of $18$ .

$\frac{12 \sqrt{3} + 4 \sqrt{9 (2)} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.3.2

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

$\frac{12 \sqrt{3} + 4 \sqrt{3^{2} \cdot 2} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.4

Pull terms out from under the radical.

$\frac{12 \sqrt{3} + 4 (3 \sqrt{2}) + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.5

Multiply $3$ by $4$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 3 \sqrt{6} \cdot 3 + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.6

Multiply $3$ by $3$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \sqrt{6} \sqrt{6}}{3}$

Step 5.1.7

Multiply $3 \sqrt{6} \sqrt{6}$ .

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Step 5.1.7.1

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 ({\sqrt{6}}^{1} \sqrt{6})}{3}$

Step 5.1.7.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{3}$

Step 5.1.7.3

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

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 {\sqrt{6}}^{1 + 1}}{3}$

Step 5.1.7.4

Add $1$ and $1$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 {\sqrt{6}}^{2}}{3}$

Step 5.1.8

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 {(6^{\frac{1}{2}})}^{2}}{3}$

Step 5.1.8.2

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

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \cdot 6^{\frac{1}{2} \cdot 2}}{3}$

Step 5.1.8.3

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

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \cdot 6^{\frac{2}{2}}}{3}$

Step 5.1.8.4

Cancel the common factor of $2$ .

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Step 5.1.8.4.1

Cancel the common factor.

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \cdot 6^{\frac{2}{2}}}{3}$

Step 5.1.8.4.2

Rewrite the expression.

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \cdot 6^{1}}{3}$

Step 5.1.8.5

Evaluate the exponent.

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 3 \cdot 6}{3}$

Step 5.1.9

Multiply $3$ by $6$ .

$\frac{12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 18}{3}$

Step 5.2

Cancel the common factor of $12 \sqrt{3} + 12 \sqrt{2} + 9 \sqrt{6} + 18$ and $3$ .

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Step 5.2.1

Factor $3$ out of $12 \sqrt{3}$ .

$\frac{3 (4 \sqrt{3}) + 12 \sqrt{2} + 9 \sqrt{6} + 18}{3}$

Step 5.2.2

Factor $3$ out of $12 \sqrt{2}$ .

$\frac{3 (4 \sqrt{3}) + 3 (4 \sqrt{2}) + 9 \sqrt{6} + 18}{3}$

Step 5.2.3

Factor $3$ out of $3 (4 \sqrt{3}) + 3 (4 \sqrt{2})$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2}) + 9 \sqrt{6} + 18}{3}$

Step 5.2.4

Factor $3$ out of $9 \sqrt{6}$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2}) + 3 (3 \sqrt{6}) + 18}{3}$

Step 5.2.5

Factor $3$ out of $3 (4 \sqrt{3} + 4 \sqrt{2}) + 3 (3 \sqrt{6})$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6}) + 18}{3}$

Step 5.2.6

Factor $3$ out of $18$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6}) + 3 \cdot 6}{3}$

Step 5.2.7

Factor $3$ out of $3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6}) + 3 (6)$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6} + 6)}{3}$

Step 5.2.8

Cancel the common factors.

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Step 5.2.8.1

Factor $3$ out of $3$ .

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6} + 6)}{3 (1)}$

Step 5.2.8.2

Cancel the common factor.

$\frac{3 (4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6} + 6)}{3 \cdot 1}$

Step 5.2.8.3

Rewrite the expression.

$\frac{4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6} + 6}{1}$

Step 5.2.8.4

Divide $4 \sqrt{3} + 4 \sqrt{2} + 3 \sqrt{6} + 6$ by $1$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$25.93352670 \dots$