Trigonometry Examples

$\sqrt{{(\sqrt{6} - 2 \sqrt{3})}^{2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 1

Rewrite ${(\sqrt{6} - 2 \sqrt{3})}^{2}$ as $(\sqrt{6} - 2 \sqrt{3}) (\sqrt{6} - 2 \sqrt{3})$ .

$\sqrt{(\sqrt{6} - 2 \sqrt{3}) (\sqrt{6} - 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 2

Expand $(\sqrt{6} - 2 \sqrt{3}) (\sqrt{6} - 2 \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{\sqrt{6} (\sqrt{6} - 2 \sqrt{3}) - 2 \sqrt{3} (\sqrt{6} - 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 2.2

Apply the distributive property.

$\sqrt{\sqrt{6} \sqrt{6} + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} (\sqrt{6} - 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 2.3

Apply the distributive property.

$\sqrt{\sqrt{6} \sqrt{6} + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

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.

$\sqrt{\sqrt{6 \cdot 6} + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.2

Multiply $6$ by $6$ .

$\sqrt{\sqrt{36} + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.3

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

$\sqrt{\sqrt{6^{2}} + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.4

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

$\sqrt{6 + \sqrt{6} (- 2 \sqrt{3}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.5

Multiply $\sqrt{6} (- 2 \sqrt{3})$ .

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Step 3.1.5.1

Combine using the product rule for radicals.

$\sqrt{6 - 2 \sqrt{6 \cdot 3} - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.5.2

Multiply $6$ by $3$ .

$\sqrt{6 - 2 \sqrt{18} - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.6

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.1.6.1

Factor $9$ out of $18$ .

$\sqrt{6 - 2 \sqrt{9 (2)} - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.6.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{6 - 2 \sqrt{3^{2} \cdot 2} - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.7

Pull terms out from under the radical.

$\sqrt{6 - 2 (3 \sqrt{2}) - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.8

Multiply $3$ by $- 2$ .

$\sqrt{6 - 6 \sqrt{2} - 2 \sqrt{3} \sqrt{6} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.9

Multiply $- 2 \sqrt{3} \sqrt{6}$ .

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Step 3.1.9.1

Combine using the product rule for radicals.

$\sqrt{6 - 6 \sqrt{2} - 2 \sqrt{6 \cdot 3} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.9.2

Multiply $6$ by $3$ .

$\sqrt{6 - 6 \sqrt{2} - 2 \sqrt{18} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.10

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.1.10.1

Factor $9$ out of $18$ .

$\sqrt{6 - 6 \sqrt{2} - 2 \sqrt{9 (2)} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.10.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{6 - 6 \sqrt{2} - 2 \sqrt{3^{2} \cdot 2} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.11

Pull terms out from under the radical.

$\sqrt{6 - 6 \sqrt{2} - 2 (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.12

Multiply $3$ by $- 2$ .

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} - 2 \sqrt{3} (- 2 \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.13

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

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Step 3.1.13.1

Multiply $- 2$ by $- 2$ .

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \sqrt{3} \sqrt{3} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.13.2

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 ({\sqrt{3}}^{1} \sqrt{3}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.13.3

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.13.4

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 {\sqrt{3}}^{1 + 1} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.13.5

Add $1$ and $1$ .

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 {\sqrt{3}}^{2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.14

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

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Step 3.1.14.1

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 {(3^{\frac{1}{2}})}^{2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.14.2

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \cdot 3^{\frac{1}{2} \cdot 2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.14.3

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

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \cdot 3^{\frac{2}{2}} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

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.

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \cdot 3^{\frac{2}{2}} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.14.4.2

Rewrite the expression.

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \cdot 3^{1} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.14.5

Evaluate the exponent.

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 4 \cdot 3 + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.1.15

Multiply $4$ by $3$ .

$\sqrt{6 - 6 \sqrt{2} - 6 \sqrt{2} + 12 + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.2

Add $6$ and $12$ .

$\sqrt{18 - 6 \sqrt{2} - 6 \sqrt{2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 3.3

Subtract $6 \sqrt{2}$ from $- 6 \sqrt{2}$ .

$\sqrt{18 - 12 \sqrt{2} + {(5 \sqrt{6} + \sqrt{3})}^{2}}$

Step 4

Rewrite ${(5 \sqrt{6} + \sqrt{3})}^{2}$ as $(5 \sqrt{6} + \sqrt{3}) (5 \sqrt{6} + \sqrt{3})$ .

$\sqrt{18 - 12 \sqrt{2} + (5 \sqrt{6} + \sqrt{3}) (5 \sqrt{6} + \sqrt{3})}$

Step 5

Expand $(5 \sqrt{6} + \sqrt{3}) (5 \sqrt{6} + \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\sqrt{18 - 12 \sqrt{2} + 5 \sqrt{6} (5 \sqrt{6} + \sqrt{3}) + \sqrt{3} (5 \sqrt{6} + \sqrt{3})}$

Step 5.2

Apply the distributive property.

$\sqrt{18 - 12 \sqrt{2} + 5 \sqrt{6} (5 \sqrt{6}) + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6} + \sqrt{3})}$

Step 5.3

Apply the distributive property.

$\sqrt{18 - 12 \sqrt{2} + 5 \sqrt{6} (5 \sqrt{6}) + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $5 \sqrt{6} (5 \sqrt{6})$ .

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Step 6.1.1.1

Multiply $5$ by $5$ .

$\sqrt{18 - 12 \sqrt{2} + 25 \sqrt{6} \sqrt{6} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.1.2

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

$\sqrt{18 - 12 \sqrt{2} + 25 ({\sqrt{6}}^{1} \sqrt{6}) + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.1.3

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

$\sqrt{18 - 12 \sqrt{2} + 25 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.1.4

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

$\sqrt{18 - 12 \sqrt{2} + 25 {\sqrt{6}}^{1 + 1} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.1.5

Add $1$ and $1$ .

$\sqrt{18 - 12 \sqrt{2} + 25 {\sqrt{6}}^{2} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2

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

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Step 6.1.2.1

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

$\sqrt{18 - 12 \sqrt{2} + 25 {(6^{\frac{1}{2}})}^{2} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2.2

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

$\sqrt{18 - 12 \sqrt{2} + 25 \cdot 6^{\frac{1}{2} \cdot 2} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2.3

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

$\sqrt{18 - 12 \sqrt{2} + 25 \cdot 6^{\frac{2}{2}} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

$\sqrt{18 - 12 \sqrt{2} + 25 \cdot 6^{\frac{2}{2}} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2.4.2

Rewrite the expression.

$\sqrt{18 - 12 \sqrt{2} + 25 \cdot 6^{1} + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.2.5

Evaluate the exponent.

$\sqrt{18 - 12 \sqrt{2} + 25 \cdot 6 + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.3

Multiply $25$ by $6$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 \sqrt{6} \sqrt{3} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.4

Multiply $5 \sqrt{6} \sqrt{3}$ .

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Step 6.1.4.1

Combine using the product rule for radicals.

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 \sqrt{3 \cdot 6} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.4.2

Multiply $3$ by $6$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 \sqrt{18} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.5

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.1.5.1

Factor $9$ out of $18$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 \sqrt{9 (2)} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.5.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 \sqrt{3^{2} \cdot 2} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.6

Pull terms out from under the radical.

$\sqrt{18 - 12 \sqrt{2} + 150 + 5 (3 \sqrt{2}) + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.7

Multiply $3$ by $5$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + \sqrt{3} (5 \sqrt{6}) + \sqrt{3} \sqrt{3}}$

Step 6.1.8

Multiply $\sqrt{3} (5 \sqrt{6})$ .

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Step 6.1.8.1

Combine using the product rule for radicals.

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 5 \sqrt{3 \cdot 6} + \sqrt{3} \sqrt{3}}$

Step 6.1.8.2

Multiply $3$ by $6$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 5 \sqrt{18} + \sqrt{3} \sqrt{3}}$

Step 6.1.9

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 6.1.9.1

Factor $9$ out of $18$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 5 \sqrt{9 (2)} + \sqrt{3} \sqrt{3}}$

Step 6.1.9.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 5 \sqrt{3^{2} \cdot 2} + \sqrt{3} \sqrt{3}}$

Step 6.1.10

Pull terms out from under the radical.

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 5 (3 \sqrt{2}) + \sqrt{3} \sqrt{3}}$

Step 6.1.11

Multiply $3$ by $5$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 15 \sqrt{2} + \sqrt{3} \sqrt{3}}$

Step 6.1.12

Combine using the product rule for radicals.

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 15 \sqrt{2} + \sqrt{3 \cdot 3}}$

Step 6.1.13

Multiply $3$ by $3$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 15 \sqrt{2} + \sqrt{9}}$

Step 6.1.14

Rewrite $9$ as $3^{2}$ .

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 15 \sqrt{2} + \sqrt{3^{2}}}$

Step 6.1.15

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

$\sqrt{18 - 12 \sqrt{2} + 150 + 15 \sqrt{2} + 15 \sqrt{2} + 3}$

Step 6.2

Add $150$ and $3$ .

$\sqrt{18 - 12 \sqrt{2} + 153 + 15 \sqrt{2} + 15 \sqrt{2}}$

Step 6.3

Add $15 \sqrt{2}$ and $15 \sqrt{2}$ .

$\sqrt{18 - 12 \sqrt{2} + 153 + 30 \sqrt{2}}$

Step 7

Add $18$ and $153$ .

$\sqrt{171 - 12 \sqrt{2} + 30 \sqrt{2}}$

Step 8

Add $- 12 \sqrt{2}$ and $30 \sqrt{2}$ .

$\sqrt{171 + 18 \sqrt{2}}$

Step 9

Rewrite $171 + 18 \sqrt{2}$ as $3^{2} (19 + 2 \sqrt{2})$ .

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Step 9.1

Factor $9$ out of $171$ .

$\sqrt{9 (19) + 18 \sqrt{2}}$

Step 9.2

Factor $9$ out of $18 \sqrt{2}$ .

$\sqrt{9 (19) + 9 (2 \sqrt{2})}$

Step 9.3

Factor $9$ out of $9 (19) + 9 (2 \sqrt{2})$ .

$\sqrt{9 (19 + 2 \sqrt{2})}$

Step 9.4

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} (19 + 2 \sqrt{2})}$

Step 10

Pull terms out from under the radical.

$3 \sqrt{19 + 2 \sqrt{2}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{19 + 2 \sqrt{2}}$

Decimal Form:

$14.01627069 \dots$