Algebra Examples

$\sqrt{2 \sqrt{7} - \sqrt{12}} \cdot \sqrt{2 \sqrt{7} + \sqrt{12}}$

Step 1

Combine using the product rule for radicals.

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

Step 2

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

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Step 2.1

Factor $4$ out of $12$ .

$\sqrt{(2 \sqrt{7} - \sqrt{4 (3)}) (2 \sqrt{7} + \sqrt{12})}$

Step 2.2

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

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

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Step 5.1

Factor $4$ out of $12$ .

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

Step 5.2

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

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

Step 6

Pull terms out from under the radical.

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

Step 7

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

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Step 7.1

Apply the distributive property.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $2 \sqrt{7} (2 \sqrt{7})$ .

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Step 8.1.1.1

Multiply $2$ by $2$ .

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

Step 8.1.1.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{4 ({\sqrt{7}}^{1} \sqrt{7}) + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.1.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{4 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.1.4

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

$\sqrt{4 {\sqrt{7}}^{1 + 1} + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.1.5

Add $1$ and $1$ .

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

Step 8.1.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{4 {(7^{\frac{1}{2}})}^{2} + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.2.2

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

$\sqrt{4 \cdot 7^{\frac{1}{2} \cdot 2} + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.2.3

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

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

Step 8.1.2.4

Cancel the common factor of $2$ .

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Step 8.1.2.4.1

Cancel the common factor.

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

Step 8.1.2.4.2

Rewrite the expression.

$\sqrt{4 \cdot 7^{1} + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.2.5

Evaluate the exponent.

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

Step 8.1.3

Multiply $4$ by $7$ .

$\sqrt{28 + 2 \sqrt{7} (2 \sqrt{3}) - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.4

Multiply $2 \sqrt{7} (2 \sqrt{3})$ .

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Step 8.1.4.1

Multiply $2$ by $2$ .

$\sqrt{28 + 4 \sqrt{7} \sqrt{3} - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.4.2

Combine using the product rule for radicals.

$\sqrt{28 + 4 \sqrt{3 \cdot 7} - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.4.3

Multiply $3$ by $7$ .

$\sqrt{28 + 4 \sqrt{21} - 2 \sqrt{3} (2 \sqrt{7}) - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.5

Multiply $- 2 \sqrt{3} (2 \sqrt{7})$ .

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Step 8.1.5.1

Multiply $2$ by $- 2$ .

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{3} \sqrt{7} - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.5.2

Combine using the product rule for radicals.

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{7 \cdot 3} - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.5.3

Multiply $7$ by $3$ .

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 2 \sqrt{3} (2 \sqrt{3})}$

Step 8.1.6

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

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Step 8.1.6.1

Multiply $2$ by $- 2$ .

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \sqrt{3} \sqrt{3}}$

Step 8.1.6.2

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 8.1.6.3

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 8.1.6.4

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 {\sqrt{3}}^{1 + 1}}$

Step 8.1.6.5

Add $1$ and $1$ .

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 {\sqrt{3}}^{2}}$

Step 8.1.7

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

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Step 8.1.7.1

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 {(3^{\frac{1}{2}})}^{2}}$

Step 8.1.7.2

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 8.1.7.3

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

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \cdot 3^{\frac{2}{2}}}$

Step 8.1.7.4

Cancel the common factor of $2$ .

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Step 8.1.7.4.1

Cancel the common factor.

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \cdot 3^{\frac{2}{2}}}$

Step 8.1.7.4.2

Rewrite the expression.

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \cdot 3^{1}}$

Step 8.1.7.5

Evaluate the exponent.

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 4 \cdot 3}$

Step 8.1.8

Multiply $- 4$ by $3$ .

$\sqrt{28 + 4 \sqrt{21} - 4 \sqrt{21} - 12}$

Step 8.2

Subtract $12$ from $28$ .

$\sqrt{16 + 4 \sqrt{21} - 4 \sqrt{21}}$

Step 8.3

Subtract $4 \sqrt{21}$ from $4 \sqrt{21}$ .

$\sqrt{16 + 0}$

Step 8.4

Add $16$ and $0$ .

$\sqrt{16}$

Step 9

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 10

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

$4$