Trigonometry Examples

$\sqrt{\frac{34 - 3 \sqrt{34}}{1 + 3 \sqrt{34}}}$

Step 1

Multiply $\frac{34 - 3 \sqrt{34}}{1 + 3 \sqrt{34}}$ by $\frac{1 - 3 \sqrt{34}}{1 - 3 \sqrt{34}}$ .

$\sqrt{\frac{34 - 3 \sqrt{34}}{1 + 3 \sqrt{34}} \cdot \frac{1 - 3 \sqrt{34}}{1 - 3 \sqrt{34}}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{34 - 3 \sqrt{34}}{1 + 3 \sqrt{34}}$ by $\frac{1 - 3 \sqrt{34}}{1 - 3 \sqrt{34}}$ .

$\sqrt{\frac{(34 - 3 \sqrt{34}) (1 - 3 \sqrt{34})}{(1 + 3 \sqrt{34}) (1 - 3 \sqrt{34})}}$

Step 2.2

Expand the denominator using the FOIL method.

$\sqrt{\frac{(34 - 3 \sqrt{34}) (1 - 3 \sqrt{34})}{1 - 3 \sqrt{34} + 3 \sqrt{34} - 9 {\sqrt{34}}^{2}}}$

Step 2.3

Simplify.

$\sqrt{\frac{(34 - 3 \sqrt{34}) (1 - 3 \sqrt{34})}{- 305}}$

Step 3

Expand $(34 - 3 \sqrt{34}) (1 - 3 \sqrt{34})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{\frac{34 (1 - 3 \sqrt{34}) - 3 \sqrt{34} (1 - 3 \sqrt{34})}{- 305}}$

Step 3.2

Apply the distributive property.

$\sqrt{\frac{34 \cdot 1 + 34 (- 3 \sqrt{34}) - 3 \sqrt{34} (1 - 3 \sqrt{34})}{- 305}}$

Step 3.3

Apply the distributive property.

$\sqrt{\frac{34 \cdot 1 + 34 (- 3 \sqrt{34}) - 3 \sqrt{34} \cdot 1 - 3 \sqrt{34} (- 3 \sqrt{34})}{- 305}}$

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 $34$ by $1$ .

$\sqrt{\frac{34 + 34 (- 3 \sqrt{34}) - 3 \sqrt{34} \cdot 1 - 3 \sqrt{34} (- 3 \sqrt{34})}{- 305}}$

Step 4.1.2

Multiply $- 3$ by $34$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} \cdot 1 - 3 \sqrt{34} (- 3 \sqrt{34})}{- 305}}$

Step 4.1.3

Multiply $- 3$ by $1$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} - 3 \sqrt{34} (- 3 \sqrt{34})}{- 305}}$

Step 4.1.4

Multiply $- 3 \sqrt{34} (- 3 \sqrt{34})$ .

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Step 4.1.4.1

Multiply $- 3$ by $- 3$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \sqrt{34} \sqrt{34}}{- 305}}$

Step 4.1.4.2

Raise $\sqrt{34}$ to the power of $1$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 ({\sqrt{34}}^{1} \sqrt{34})}{- 305}}$

Step 4.1.4.3

Raise $\sqrt{34}$ to the power of $1$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 ({\sqrt{34}}^{1} {\sqrt{34}}^{1})}{- 305}}$

Step 4.1.4.4

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

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 {\sqrt{34}}^{1 + 1}}{- 305}}$

Step 4.1.4.5

Add $1$ and $1$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 {\sqrt{34}}^{2}}{- 305}}$

Step 4.1.5

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 {(34^{\frac{1}{2}})}^{2}}{- 305}}$

Step 4.1.5.2

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

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \cdot 34^{\frac{1}{2} \cdot 2}}{- 305}}$

Step 4.1.5.3

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

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \cdot 34^{\frac{2}{2}}}{- 305}}$

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.

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \cdot 34^{\frac{2}{2}}}{- 305}}$

Step 4.1.5.4.2

Rewrite the expression.

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \cdot 34^{1}}{- 305}}$

Step 4.1.5.5

Evaluate the exponent.

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 9 \cdot 34}{- 305}}$

Step 4.1.6

Multiply $9$ by $34$ .

$\sqrt{\frac{34 - 102 \sqrt{34} - 3 \sqrt{34} + 306}{- 305}}$

Step 4.2

Add $34$ and $306$ .

$\sqrt{\frac{340 - 102 \sqrt{34} - 3 \sqrt{34}}{- 305}}$

Step 4.3

Subtract $3 \sqrt{34}$ from $- 102 \sqrt{34}$ .

$\sqrt{\frac{340 - 105 \sqrt{34}}{- 305}}$

Step 5

Cancel the common factor of $340 - 105 \sqrt{34}$ and $- 305$ .

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Step 5.1

Factor $5$ out of $340$ .

$\sqrt{\frac{5 (68) - 105 \sqrt{34}}{- 305}}$

Step 5.2

Factor $5$ out of $- 105 \sqrt{34}$ .

$\sqrt{\frac{5 (68) + 5 (- 21 \sqrt{34})}{- 305}}$

Step 5.3

Factor $5$ out of $5 (68) + 5 (- 21 \sqrt{34})$ .

$\sqrt{\frac{5 (68 - 21 \sqrt{34})}{- 305}}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $5$ out of $- 305$ .

$\sqrt{\frac{5 (68 - 21 \sqrt{34})}{5 \cdot - 61}}$

Step 5.4.2

Cancel the common factor.

$\sqrt{\frac{5 (68 - 21 \sqrt{34})}{5 \cdot - 61}}$

Step 5.4.3

Rewrite the expression.

$\sqrt{\frac{68 - 21 \sqrt{34}}{- 61}}$

Step 6

Simplify the expression.

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Step 6.1

Move the negative in front of the fraction.

$\sqrt{- \frac{68 - 21 \sqrt{34}}{61}}$

Step 6.2

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{68 - 21 \sqrt{34}}{61}}$

Step 7

Pull terms out from under the radical.

$i \sqrt{\frac{68 - 21 \sqrt{34}}{61}}$

Step 8

Rewrite $\sqrt{\frac{68 - 21 \sqrt{34}}{61}}$ as $\frac{\sqrt{68 - 21 \sqrt{34}}}{\sqrt{61}}$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}}}{\sqrt{61}}$

Step 9

Multiply $\frac{\sqrt{68 - 21 \sqrt{34}}}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$i (\frac{\sqrt{68 - 21 \sqrt{34}}}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}})$

Step 10

Simplify terms.

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Step 10.1

Combine and simplify the denominator.

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Step 10.1.1

Multiply $\frac{\sqrt{68 - 21 \sqrt{34}}}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{\sqrt{61} \sqrt{61}}$

Step 10.1.2

Raise $\sqrt{61}$ to the power of $1$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{{\sqrt{61}}^{1} \sqrt{61}}$

Step 10.1.3

Raise $\sqrt{61}$ to the power of $1$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{{\sqrt{61}}^{1} {\sqrt{61}}^{1}}$

Step 10.1.4

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

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{{\sqrt{61}}^{1 + 1}}$

Step 10.1.5

Add $1$ and $1$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{{\sqrt{61}}^{2}}$

Step 10.1.6

Rewrite ${\sqrt{61}}^{2}$ as $61$ .

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Step 10.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{61}$ as $61^{\frac{1}{2}}$ .

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{{(61^{\frac{1}{2}})}^{2}}$

Step 10.1.6.2

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

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{61^{\frac{1}{2} \cdot 2}}$

Step 10.1.6.3

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

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{61^{\frac{2}{2}}}$

Step 10.1.6.4

Cancel the common factor of $2$ .

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Step 10.1.6.4.1

Cancel the common factor.

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{61^{\frac{2}{2}}}$

Step 10.1.6.4.2

Rewrite the expression.

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{61^{1}}$

Step 10.1.6.5

Evaluate the exponent.

$i \frac{\sqrt{68 - 21 \sqrt{34}} \sqrt{61}}{61}$

Step 10.2

Combine using the product rule for radicals.

$i \frac{\sqrt{(68 - 21 \sqrt{34}) \cdot 61}}{61}$

Step 10.3

Combine $i$ and $\frac{\sqrt{(68 - 21 \sqrt{34}) \cdot 61}}{61}$ .

$\frac{i \sqrt{(68 - 21 \sqrt{34}) \cdot 61}}{61}$

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

$\frac{i \sqrt{68 \cdot 61 - 21 \sqrt{34} \cdot 61}}{61}$

Step 11.2

Multiply $68$ by $61$ .

$\frac{i \sqrt{4148 - 21 \sqrt{34} \cdot 61}}{61}$

Step 11.3

Multiply $61$ by $- 21$ .

$\frac{i \sqrt{4148 - 1281 \sqrt{34}}}{61}$