Algebra Examples

$(\sqrt{14} - \sqrt{3}) (\sqrt{12} + \sqrt{7})$

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 $12$ .

$(\sqrt{14} - \sqrt{3}) (\sqrt{4 (3)} + \sqrt{7})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Expand $(\sqrt{14} - \sqrt{3}) (2 \sqrt{3} + \sqrt{7})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $\sqrt{14} (2 \sqrt{3})$ .

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Step 3.1.1

Combine using the product rule for radicals.

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

Step 3.1.2

Multiply $14$ by $3$ .

$2 \sqrt{42} + \sqrt{14} \sqrt{7} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.2

Combine using the product rule for radicals.

$2 \sqrt{42} + \sqrt{14 \cdot 7} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.3

Multiply $14$ by $7$ .

$2 \sqrt{42} + \sqrt{98} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.4

Rewrite $98$ as $7^{2} \cdot 2$ .

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Step 3.4.1

Factor $49$ out of $98$ .

$2 \sqrt{42} + \sqrt{49 (2)} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.4.2

Rewrite $49$ as $7^{2}$ .

$2 \sqrt{42} + \sqrt{7^{2} \cdot 2} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.5

Pull terms out from under the radical.

$2 \sqrt{42} + 7 \sqrt{2} - \sqrt{3} (2 \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.6

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

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Step 3.6.1

Multiply $2$ by $- 1$ .

$2 \sqrt{42} + 7 \sqrt{2} - 2 \sqrt{3} \sqrt{3} - \sqrt{3} \sqrt{7}$

Step 3.6.2

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 ({\sqrt{3}}^{1} \sqrt{3}) - \sqrt{3} \sqrt{7}$

Step 3.6.3

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - \sqrt{3} \sqrt{7}$

Step 3.6.4

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 {\sqrt{3}}^{1 + 1} - \sqrt{3} \sqrt{7}$

Step 3.6.5

Add $1$ and $1$ .

$2 \sqrt{42} + 7 \sqrt{2} - 2 {\sqrt{3}}^{2} - \sqrt{3} \sqrt{7}$

Step 3.7

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

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Step 3.7.1

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 {(3^{\frac{1}{2}})}^{2} - \sqrt{3} \sqrt{7}$

Step 3.7.2

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 \cdot 3^{\frac{1}{2} \cdot 2} - \sqrt{3} \sqrt{7}$

Step 3.7.3

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

$2 \sqrt{42} + 7 \sqrt{2} - 2 \cdot 3^{\frac{2}{2}} - \sqrt{3} \sqrt{7}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$2 \sqrt{42} + 7 \sqrt{2} - 2 \cdot 3^{\frac{2}{2}} - \sqrt{3} \sqrt{7}$

Step 3.7.4.2

Rewrite the expression.

$2 \sqrt{42} + 7 \sqrt{2} - 2 \cdot 3^{1} - \sqrt{3} \sqrt{7}$

Step 3.7.5

Evaluate the exponent.

$2 \sqrt{42} + 7 \sqrt{2} - 2 \cdot 3 - \sqrt{3} \sqrt{7}$

Step 3.8

Multiply $- 2$ by $3$ .

$2 \sqrt{42} + 7 \sqrt{2} - 6 - \sqrt{3} \sqrt{7}$

Step 3.9

Multiply $- \sqrt{3} \sqrt{7}$ .

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Step 3.9.1

Combine using the product rule for radicals.

$2 \sqrt{42} + 7 \sqrt{2} - 6 - \sqrt{7 \cdot 3}$

Step 3.9.2

Multiply $7$ by $3$ .

$2 \sqrt{42} + 7 \sqrt{2} - 6 - \sqrt{21}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{42} + 7 \sqrt{2} - 6 - \sqrt{21}$

Decimal Form:

$12.27840063 \dots$