Basic Math Examples

$(\sqrt{6} - \sqrt{8}) (\sqrt{6} + \sqrt{8})$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1.1.1

Factor $4$ out of $8$ .

$(\sqrt{6} - \sqrt{4 (2)}) (\sqrt{6} + \sqrt{8})$

Step 1.1.1.2

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

$(\sqrt{6} - \sqrt{2^{2} \cdot 2}) (\sqrt{6} + \sqrt{8})$

Step 1.1.2

Pull terms out from under the radical.

$(\sqrt{6} - (2 \sqrt{2})) (\sqrt{6} + \sqrt{8})$

Step 1.1.3

Multiply $2$ by $- 1$ .

$(\sqrt{6} - 2 \sqrt{2}) (\sqrt{6} + \sqrt{8})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.2.1.1

Factor $4$ out of $8$ .

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

Step 1.2.1.2

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

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

Step 1.2.2

Pull terms out from under the radical.

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $6$ by $6$ .

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

Step 3.1.3

Rewrite $36$ as $6^{2}$ .

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

Step 3.1.4

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3.1.5

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

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Step 3.1.5.1

Combine using the product rule for radicals.

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

Step 3.1.5.2

Multiply $6$ by $2$ .

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

Step 3.1.6

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

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Step 3.1.6.1

Factor $4$ out of $12$ .

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

Step 3.1.6.2

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

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

Step 3.1.7

Pull terms out from under the radical.

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

Step 3.1.8

Multiply $2$ by $2$ .

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

Step 3.1.9

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

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Step 3.1.9.1

Combine using the product rule for radicals.

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

Step 3.1.9.2

Multiply $6$ by $2$ .

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

Step 3.1.10

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

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Step 3.1.10.1

Factor $4$ out of $12$ .

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

Step 3.1.10.2

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

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

Step 3.1.11

Pull terms out from under the radical.

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

Step 3.1.12

Multiply $2$ by $- 2$ .

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

Step 3.1.13

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

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Step 3.1.13.1

Multiply $2$ by $- 2$ .

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

Step 3.1.13.2

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

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

Step 3.1.13.3

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

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

Step 3.1.13.4

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

$6 + 4 \sqrt{3} - 4 \sqrt{3} - 4 {\sqrt{2}}^{1 + 1}$

Step 3.1.13.5

Add $1$ and $1$ .

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

Step 3.1.14

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

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Step 3.1.14.1

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

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

Step 3.1.14.2

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

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

Step 3.1.14.3

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

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

Step 3.1.14.4

Cancel the common factor of $2$ .

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Step 3.1.14.4.1

Cancel the common factor.

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

Step 3.1.14.4.2

Rewrite the expression.

$6 + 4 \sqrt{3} - 4 \sqrt{3} - 4 \cdot 2^{1}$

Step 3.1.14.5

Evaluate the exponent.

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

Step 3.1.15

Multiply $- 4$ by $2$ .

$6 + 4 \sqrt{3} - 4 \sqrt{3} - 8$

Step 3.2

Subtract $8$ from $6$ .

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

Step 3.3

Subtract $4 \sqrt{3}$ from $4 \sqrt{3}$ .

$- 2 + 0$

Step 3.4

Add $- 2$ and $0$ .

$- 2$