Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $6 \sqrt{6} \sqrt{6}$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

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.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Evaluate the exponent.

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

Step 2.1.3

Multiply $6$ by $6$ .

$36 + 6 \sqrt{6} \sqrt{2} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.4

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

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Step 2.1.4.1

Combine using the product rule for radicals.

$36 + 6 \sqrt{2 \cdot 6} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.4.2

Multiply $2$ by $6$ .

$36 + 6 \sqrt{12} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.5

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

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Step 2.1.5.1

Factor $4$ out of $12$ .

$36 + 6 \sqrt{4 (3)} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.5.2

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

$36 + 6 \sqrt{2^{2} \cdot 3} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.6

Pull terms out from under the radical.

$36 + 6 (2 \sqrt{3}) - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.7

Multiply $2$ by $6$ .

$36 + 12 \sqrt{3} - 6 \sqrt{2} \sqrt{6} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.8

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

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Step 2.1.8.1

Combine using the product rule for radicals.

$36 + 12 \sqrt{3} - 6 \sqrt{6 \cdot 2} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.8.2

Multiply $6$ by $2$ .

$36 + 12 \sqrt{3} - 6 \sqrt{12} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.9

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

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Step 2.1.9.1

Factor $4$ out of $12$ .

$36 + 12 \sqrt{3} - 6 \sqrt{4 (3)} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.9.2

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

$36 + 12 \sqrt{3} - 6 \sqrt{2^{2} \cdot 3} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.10

Pull terms out from under the radical.

$36 + 12 \sqrt{3} - 6 (2 \sqrt{3}) - 6 \sqrt{2} \sqrt{2}$

Step 2.1.11

Multiply $2$ by $- 6$ .

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \sqrt{2} \sqrt{2}$

Step 2.1.12

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

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Step 2.1.12.1

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 ({\sqrt{2}}^{1} \sqrt{2})$

Step 2.1.12.2

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 2.1.12.3

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 {\sqrt{2}}^{1 + 1}$

Step 2.1.12.4

Add $1$ and $1$ .

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 {\sqrt{2}}^{2}$

Step 2.1.13

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

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Step 2.1.13.1

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 {(2^{\frac{1}{2}})}^{2}$

Step 2.1.13.2

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 2.1.13.3

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

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \cdot 2^{\frac{2}{2}}$

Step 2.1.13.4

Cancel the common factor of $2$ .

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Step 2.1.13.4.1

Cancel the common factor.

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \cdot 2^{\frac{2}{2}}$

Step 2.1.13.4.2

Rewrite the expression.

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \cdot 2^{1}$

Step 2.1.13.5

Evaluate the exponent.

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 6 \cdot 2$

Step 2.1.14

Multiply $- 6$ by $2$ .

$36 + 12 \sqrt{3} - 12 \sqrt{3} - 12$

Step 2.2

Subtract $12$ from $36$ .

$24 + 12 \sqrt{3} - 12 \sqrt{3}$

Step 2.3

Subtract $12 \sqrt{3}$ from $12 \sqrt{3}$ .

$24 + 0$

Step 2.4

Add $24$ and $0$ .

$24$