Algebra Examples

$(7 + \sqrt{12}) (7 - \sqrt{12})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$(7 + \sqrt{4 (3)}) (7 - \sqrt{12})$

Step 1.1.2

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

$(7 + \sqrt{2^{2} \cdot 3}) (7 - \sqrt{12})$

Step 1.2

Pull terms out from under the radical.

$(7 + | 2 | \sqrt{3}) (7 - \sqrt{12})$

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$(7 + 2 \sqrt{3}) (7 - \sqrt{12})$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $12$ .

$(7 + 2 \sqrt{3}) (7 - \sqrt{4 (3)})$

Step 2.1.2

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

$(7 + 2 \sqrt{3}) (7 - \sqrt{2^{2} \cdot 3})$

Step 2.2

Pull terms out from under the radical.

$(7 + 2 \sqrt{3}) (7 - (| 2 | \sqrt{3}))$

Step 2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$(7 + 2 \sqrt{3}) (7 - (2 \sqrt{3}))$

Step 2.4

Multiply $2$ by $- 1$ .

$(7 + 2 \sqrt{3}) (7 - 2 \sqrt{3})$

Step 3

Expand $(7 + 2 \sqrt{3}) (7 - 2 \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$7 (7 - 2 \sqrt{3}) + 2 \sqrt{3} (7 - 2 \sqrt{3})$

Step 3.2

Apply the distributive property.

$7 \cdot 7 + 7 (- 2 \sqrt{3}) + 2 \sqrt{3} (7 - 2 \sqrt{3})$

Step 3.3

Apply the distributive property.

$7 \cdot 7 + 7 (- 2 \sqrt{3}) + 2 \sqrt{3} \cdot 7 + 2 \sqrt{3} (- 2 \sqrt{3})$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $7$ by $7$ .

$49 + 7 (- 2 \sqrt{3}) + 2 \sqrt{3} \cdot 7 + 2 \sqrt{3} (- 2 \sqrt{3})$

Step 4.1.2

Multiply $- 2$ by $7$ .

$49 - 14 \sqrt{3} + 2 \sqrt{3} \cdot 7 + 2 \sqrt{3} (- 2 \sqrt{3})$

Step 4.1.3

Multiply $7$ by $2$ .

$49 - 14 \sqrt{3} + 14 \sqrt{3} + 2 \sqrt{3} (- 2 \sqrt{3})$

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 2$ by $2$ .

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \sqrt{3} \sqrt{3}$

Step 4.1.4.2

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 ({\sqrt{3}}^{1} \sqrt{3})$

Step 4.1.4.3

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 4.1.4.4

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 {\sqrt{3}}^{1 + 1}$

Step 4.1.4.5

Add $1$ and $1$ .

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 {\sqrt{3}}^{2}$

Step 4.1.5

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

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Step 4.1.5.1

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}$

Step 4.1.5.2

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 4.1.5.3

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

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Step 4.1.5.4.2

Rewrite the expression.

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \cdot 3^{1}$

Step 4.1.5.5

Evaluate the exponent.

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 4 \cdot 3$

Step 4.1.6

Multiply $- 4$ by $3$ .

$49 - 14 \sqrt{3} + 14 \sqrt{3} - 12$

Step 4.2

Subtract $12$ from $49$ .

$37 - 14 \sqrt{3} + 14 \sqrt{3}$

Step 4.3

Add $- 14 \sqrt{3}$ and $14 \sqrt{3}$ .

$37 + 0$

Step 4.4

Add $37$ and $0$ .

$37$