Trigonometry Examples

$\frac{4 - 6 \sqrt{3 n}}{3 \sqrt{11 n}}$

Step 1

Factor $2$ out of $4 - 6 \sqrt{3 n}$ .

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Step 1.1

Factor $2$ out of $4$ .

$\frac{2 (2) - 6 \sqrt{3 n}}{3 \sqrt{11 n}}$

Step 1.2

Factor $2$ out of $- 6 \sqrt{3 n}$ .

$\frac{2 (2) + 2 (- 3 \sqrt{3 n})}{3 \sqrt{11 n}}$

Step 1.3

Factor $2$ out of $2 (2) + 2 (- 3 \sqrt{3 n})$ .

$\frac{2 (2 - 3 \sqrt{3 n})}{3 \sqrt{11 n}}$

Step 2

Multiply $\frac{2 (2 - 3 \sqrt{3 n})}{3 \sqrt{11 n}}$ by $\frac{\sqrt{11 n}}{\sqrt{11 n}}$ .

$\frac{2 (2 - 3 \sqrt{3 n})}{3 \sqrt{11 n}} \cdot \frac{\sqrt{11 n}}{\sqrt{11 n}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{2 (2 - 3 \sqrt{3 n})}{3 \sqrt{11 n}}$ by $\frac{\sqrt{11 n}}{\sqrt{11 n}}$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 \sqrt{11 n} \sqrt{11 n}}$

Step 3.1.2

Move $\sqrt{11 n}$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 (\sqrt{11 n} \sqrt{11 n})}$

Step 3.1.3

Raise $\sqrt{11 n}$ to the power of $1$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 ({\sqrt{11 n}}^{1} \sqrt{11 n})}$

Step 3.1.4

Raise $\sqrt{11 n}$ to the power of $1$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 ({\sqrt{11 n}}^{1} {\sqrt{11 n}}^{1})}$

Step 3.1.5

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

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {\sqrt{11 n}}^{1 + 1}}$

Step 3.1.6

Add $1$ and $1$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {\sqrt{11 n}}^{2}}$

Step 3.1.7

Rewrite ${\sqrt{11 n}}^{2}$ as $11 n$ .

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Step 3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 n}$ as ${(11 n)}^{\frac{1}{2}}$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {({(11 n)}^{\frac{1}{2}})}^{2}}$

Step 3.1.7.2

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

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {(11 n)}^{\frac{1}{2} \cdot 2}}$

Step 3.1.7.3

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

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {(11 n)}^{\frac{2}{2}}}$

Step 3.1.7.4

Cancel the common factor of $2$ .

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Step 3.1.7.4.1

Cancel the common factor.

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {(11 n)}^{\frac{2}{2}}}$

Step 3.1.7.4.2

Rewrite the expression.

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 {(11 n)}^{1}}$

Step 3.1.7.5

Simplify.

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{3 (11 n)}$

Step 3.2

Simplify by multiplying through.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Multiply $3$ by $11$ .

$\frac{2 (2 - 3 \sqrt{3 n}) \sqrt{11 n}}{33 n}$

Step 3.2.1.2

Group $\sqrt{11 n}$ and $2 - 3 \sqrt{3 n}$ together.

$\frac{2 (\sqrt{11 n} (2 - 3 \sqrt{3 n}))}{33 n}$

Step 3.2.2

Apply the distributive property.

$\frac{2 (\sqrt{11 n} \cdot 2 + \sqrt{11 n} (- 3 \sqrt{3 n}))}{33 n}$

Step 3.2.3

Reorder.

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Step 3.2.3.1

Move $2$ to the left of $\sqrt{11 n}$ .

$\frac{2 (2 \cdot \sqrt{11 n} + \sqrt{11 n} (- 3 \sqrt{3 n}))}{33 n}$

Step 3.2.3.2

Rewrite using the commutative property of multiplication.

$\frac{2 (2 \cdot \sqrt{11 n} - 3 \sqrt{11 n} \sqrt{3 n})}{33 n}$

Step 3.3

Simplify each term.

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Step 3.3.1

Multiply $- 3 \sqrt{11 n} \sqrt{3 n}$ .

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Step 3.3.1.1

Combine using the product rule for radicals.

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{3 n (11 n)})}{33 n}$

Step 3.3.1.2

Multiply $11$ by $3$ .

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{33 n \cdot n})}{33 n}$

Step 3.3.1.3

Raise $n$ to the power of $1$ .

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{33 (n^{1} n)})}{33 n}$

Step 3.3.1.4

Raise $n$ to the power of $1$ .

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{33 (n^{1} n^{1})})}{33 n}$

Step 3.3.1.5

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

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{33 n^{1 + 1}})}{33 n}$

Step 3.3.1.6

Add $1$ and $1$ .

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{33 n^{2}})}{33 n}$

Step 3.3.2

Reorder $33$ and $n^{2}$ .

$\frac{2 (2 \sqrt{11 n} - 3 \sqrt{n^{2} \cdot 33})}{33 n}$

Step 3.3.3

Pull terms out from under the radical.

$\frac{2 (2 \sqrt{11 n} - 3 n \sqrt{33})}{33 n}$

Step 4

Simplify the numerator.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 n}$ as ${(11 n)}^{\frac{1}{2}}$ .

$\frac{2 (2 {(11 n)}^{\frac{1}{2}} - 3 n \sqrt{33})}{33 n}$

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\frac{2 (2 {(11 n)}^{\frac{1}{2}} - 3 n \cdot 33^{\frac{1}{2}})}{33 n}$

Step 4.3

Apply the product rule to $11 n$ .

$\frac{2 (2 (11^{\frac{1}{2}} n^{\frac{1}{2}}) - 3 n \cdot 33^{\frac{1}{2}})}{33 n}$

Step 4.4

Factor $n^{\frac{1}{2}}$ out of $2 \cdot 11^{\frac{1}{2}} n^{\frac{1}{2}} - 3 n \cdot 33^{\frac{1}{2}}$ .

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Step 4.4.1

Move $n$ .

$\frac{2 (2 \cdot 11^{\frac{1}{2}} n^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n)}{33 n}$

Step 4.4.2

Factor $n^{\frac{1}{2}}$ out of $2 \cdot 11^{\frac{1}{2}} n^{\frac{1}{2}}$ .

$\frac{2 (n^{\frac{1}{2}} (2 \cdot 11^{\frac{1}{2}}) - 3 \cdot 33^{\frac{1}{2}} n)}{33 n}$

Step 4.4.3

Factor $n^{\frac{1}{2}}$ out of $- 3 \cdot 33^{\frac{1}{2}} n$ .

$\frac{2 (n^{\frac{1}{2}} (2 \cdot 11^{\frac{1}{2}}) + n^{\frac{1}{2}} (- 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}}))}{33 n}$

Step 4.4.4

Factor $n^{\frac{1}{2}}$ out of $n^{\frac{1}{2}} (2 \cdot 11^{\frac{1}{2}}) + n^{\frac{1}{2}} (- 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})$ .

$\frac{2 (n^{\frac{1}{2}} (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}}))}{33 n}$

$\frac{2 n^{\frac{1}{2}} (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n}$

Step 5

Move $n^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n \cdot n^{- \frac{1}{2}}}$

Step 6

Multiply $n$ by $n^{- \frac{1}{2}}$ by adding the exponents.

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Step 6.1

Move $n^{- \frac{1}{2}}$ .

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 (n^{- \frac{1}{2}} n)}$

Step 6.2

Multiply $n^{- \frac{1}{2}}$ by $n$ .

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Step 6.2.1

Raise $n$ to the power of $1$ .

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 (n^{- \frac{1}{2}} n^{1})}$

Step 6.2.2

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

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n^{- \frac{1}{2} + 1}}$

Step 6.3

Write $1$ as a fraction with a common denominator.

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n^{- \frac{1}{2} + \frac{2}{2}}}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n^{\frac{- 1 + 2}{2}}}$

Step 6.5

Add $- 1$ and $2$ .

$\frac{2 (2 \cdot 11^{\frac{1}{2}} - 3 \cdot 33^{\frac{1}{2}} n^{\frac{1}{2}})}{33 n^{\frac{1}{2}}}$