Trigonometry Examples

$\frac{40 \sqrt{10} - 63}{203} \div \frac{- 42 \sqrt{10} + 60}{203}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{40 \sqrt{10} - 63}{203} \cdot \frac{203}{- 42 \sqrt{10} + 60}$

Step 2

Cancel the common factor of $203$ .

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Step 2.1

Cancel the common factor.

$\frac{40 \sqrt{10} - 63}{203} \cdot \frac{203}{- 42 \sqrt{10} + 60}$

Step 2.2

Rewrite the expression.

$(40 \sqrt{10} - 63) \frac{1}{- 42 \sqrt{10} + 60}$

Step 3

Multiply $\frac{1}{- 42 \sqrt{10} + 60}$ by $\frac{- 42 \sqrt{10} - 60}{- 42 \sqrt{10} - 60}$ .

$(40 \sqrt{10} - 63) (\frac{1}{- 42 \sqrt{10} + 60} \cdot \frac{- 42 \sqrt{10} - 60}{- 42 \sqrt{10} - 60})$

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{1}{- 42 \sqrt{10} + 60}$ by $\frac{- 42 \sqrt{10} - 60}{- 42 \sqrt{10} - 60}$ .

$(40 \sqrt{10} - 63) \frac{- 42 \sqrt{10} - 60}{(- 42 \sqrt{10} + 60) (- 42 \sqrt{10} - 60)}$

Step 4.2

Expand the denominator using the FOIL method.

$(40 \sqrt{10} - 63) \frac{- 42 \sqrt{10} - 60}{1764 {\sqrt{10}}^{2} + 2520 \sqrt{10} - 2520 \sqrt{10} - 3600}$

Step 4.3

Simplify.

$(40 \sqrt{10} - 63) \frac{- 42 \sqrt{10} - 60}{14040}$

Step 4.4

Cancel the common factor of $- 42 \sqrt{10} - 60$ and $14040$ .

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Step 4.4.1

Factor $6$ out of $- 42 \sqrt{10}$ .

$(40 \sqrt{10} - 63) \frac{6 (- 7 \sqrt{10}) - 60}{14040}$

Step 4.4.2

Factor $6$ out of $- 60$ .

$(40 \sqrt{10} - 63) \frac{6 (- 7 \sqrt{10}) + 6 (- 10)}{14040}$

Step 4.4.3

Factor $6$ out of $6 (- 7 \sqrt{10}) + 6 (- 10)$ .

$(40 \sqrt{10} - 63) \frac{6 (- 7 \sqrt{10} - 10)}{14040}$

Step 4.4.4

Cancel the common factors.

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Step 4.4.4.1

Factor $6$ out of $14040$ .

$(40 \sqrt{10} - 63) \frac{6 (- 7 \sqrt{10} - 10)}{6 (2340)}$

Step 4.4.4.2

Cancel the common factor.

$(40 \sqrt{10} - 63) \frac{6 (- 7 \sqrt{10} - 10)}{6 \cdot 2340}$

Step 4.4.4.3

Rewrite the expression.

$(40 \sqrt{10} - 63) \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.5

Apply the distributive property.

$40 \sqrt{10} \frac{- 7 \sqrt{10} - 10}{2340} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.6

Cancel the common factor of $20$ .

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Step 4.6.1

Factor $20$ out of $40 \sqrt{10}$ .

$20 (2 \sqrt{10}) \frac{- 7 \sqrt{10} - 10}{2340} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.6.2

Factor $20$ out of $2340$ .

$20 (2 \sqrt{10}) \frac{- 7 \sqrt{10} - 10}{20 (117)} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.6.3

Cancel the common factor.

$20 (2 \sqrt{10}) \frac{- 7 \sqrt{10} - 10}{20 \cdot 117} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.6.4

Rewrite the expression.

$2 \sqrt{10} \frac{- 7 \sqrt{10} - 10}{117} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.7

Combine $\frac{- 7 \sqrt{10} - 10}{117}$ and $2$ .

$\frac{(- 7 \sqrt{10} - 10) \cdot 2}{117} \sqrt{10} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.8

Combine $\frac{(- 7 \sqrt{10} - 10) \cdot 2}{117}$ and $\sqrt{10}$ .

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} - 63 \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.9

Cancel the common factor of $9$ .

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Step 4.9.1

Factor $9$ out of $- 63$ .

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} + 9 (- 7) \frac{- 7 \sqrt{10} - 10}{2340}$

Step 4.9.2

Factor $9$ out of $2340$ .

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} + 9 \cdot - 7 \frac{- 7 \sqrt{10} - 10}{9 \cdot 260}$

Step 4.9.3

Cancel the common factor.

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} + 9 \cdot - 7 \frac{- 7 \sqrt{10} - 10}{9 \cdot 260}$

Step 4.9.4

Rewrite the expression.

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} - 7 \frac{- 7 \sqrt{10} - 10}{260}$

Step 4.10

Combine $- 7$ and $\frac{- 7 \sqrt{10} - 10}{260}$ .

$\frac{(- 7 \sqrt{10} - 10) \cdot 2 \sqrt{10}}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5

Simplify each term.

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Step 5.1

Group $\sqrt{10}$ and $- 7 \sqrt{10} - 10$ together.

$\frac{\sqrt{10} (- 7 \sqrt{10} - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.2

Apply the distributive property.

$\frac{(\sqrt{10} (- 7 \sqrt{10}) + \sqrt{10} \cdot - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.3

Multiply $\sqrt{10} (- 7 \sqrt{10})$ .

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Step 5.3.1

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{(- 7 ({\sqrt{10}}^{1} \sqrt{10}) + \sqrt{10} \cdot - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{(- 7 ({\sqrt{10}}^{1} {\sqrt{10}}^{1}) + \sqrt{10} \cdot - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.3.3

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

$\frac{(- 7 {\sqrt{10}}^{1 + 1} + \sqrt{10} \cdot - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.3.4

Add $1$ and $1$ .

$\frac{(- 7 {\sqrt{10}}^{2} + \sqrt{10} \cdot - 10) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.4

Move $- 10$ to the left of $\sqrt{10}$ .

$\frac{(- 7 {\sqrt{10}}^{2} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5

Simplify each term.

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Step 5.5.1

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 5.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{(- 7 {(10^{\frac{1}{2}})}^{2} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.1.2

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

$\frac{(- 7 \cdot 10^{\frac{1}{2} \cdot 2} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.1.3

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

$\frac{(- 7 \cdot 10^{\frac{2}{2}} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.1.4

Cancel the common factor of $2$ .

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Step 5.5.1.4.1

Cancel the common factor.

$\frac{(- 7 \cdot 10^{\frac{2}{2}} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.1.4.2

Rewrite the expression.

$\frac{(- 7 \cdot 10^{1} - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.1.5

Evaluate the exponent.

$\frac{(- 7 \cdot 10 - 10 \cdot \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.5.2

Multiply $- 7$ by $10$ .

$\frac{(- 70 - 10 \sqrt{10}) \cdot 2}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.6

Move $2$ to the left of $- 70 - 10 \sqrt{10}$ .

$\frac{2 \cdot (- 70 - 10 \sqrt{10})}{117} + \frac{- 7 (- 7 \sqrt{10} - 10)}{260}$

Step 5.7

Move the negative in front of the fraction.

$\frac{2 (- 70 - 10 \sqrt{10})}{117} - \frac{7 (- 7 \sqrt{10} - 10)}{260}$

Step 6

To write $\frac{2 (- 70 - 10 \sqrt{10})}{117}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{2 (- 70 - 10 \sqrt{10})}{117} \cdot \frac{20}{20} - \frac{7 (- 7 \sqrt{10} - 10)}{260}$

Step 7

To write $- \frac{7 (- 7 \sqrt{10} - 10)}{260}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 (- 70 - 10 \sqrt{10})}{117} \cdot \frac{20}{20} - \frac{7 (- 7 \sqrt{10} - 10)}{260} \cdot \frac{9}{9}$

Step 8

Write each expression with a common denominator of $2340$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{2 (- 70 - 10 \sqrt{10})}{117}$ by $\frac{20}{20}$ .

$\frac{2 (- 70 - 10 \sqrt{10}) \cdot 20}{117 \cdot 20} - \frac{7 (- 7 \sqrt{10} - 10)}{260} \cdot \frac{9}{9}$

Step 8.2

Multiply $117$ by $20$ .

$\frac{2 (- 70 - 10 \sqrt{10}) \cdot 20}{2340} - \frac{7 (- 7 \sqrt{10} - 10)}{260} \cdot \frac{9}{9}$

Step 8.3

Multiply $\frac{7 (- 7 \sqrt{10} - 10)}{260}$ by $\frac{9}{9}$ .

$\frac{2 (- 70 - 10 \sqrt{10}) \cdot 20}{2340} - \frac{7 (- 7 \sqrt{10} - 10) \cdot 9}{260 \cdot 9}$

Step 8.4

Multiply $260$ by $9$ .

$\frac{2 (- 70 - 10 \sqrt{10}) \cdot 20}{2340} - \frac{7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 9

Combine the numerators over the common denominator.

$\frac{2 (- 70 - 10 \sqrt{10}) \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{(2 \cdot - 70 + 2 (- 10 \sqrt{10})) \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.2

Multiply $2$ by $- 70$ .

$\frac{(- 140 + 2 (- 10 \sqrt{10})) \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.3

Multiply $- 10$ by $2$ .

$\frac{(- 140 - 20 \sqrt{10}) \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.4

Apply the distributive property.

$\frac{- 140 \cdot 20 - 20 \sqrt{10} \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.5

Multiply $- 140$ by $20$ .

$\frac{- 2800 - 20 \sqrt{10} \cdot 20 - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.6

Multiply $20$ by $- 20$ .

$\frac{- 2800 - 400 \sqrt{10} - 7 (- 7 \sqrt{10} - 10) \cdot 9}{2340}$

Step 10.7

Apply the distributive property.

$\frac{- 2800 - 400 \sqrt{10} + (- 7 (- 7 \sqrt{10}) - 7 \cdot - 10) \cdot 9}{2340}$

Step 10.8

Multiply $- 7$ by $- 7$ .

$\frac{- 2800 - 400 \sqrt{10} + (49 \sqrt{10} - 7 \cdot - 10) \cdot 9}{2340}$

Step 10.9

Multiply $- 7$ by $- 10$ .

$\frac{- 2800 - 400 \sqrt{10} + (49 \sqrt{10} + 70) \cdot 9}{2340}$

Step 10.10

Apply the distributive property.

$\frac{- 2800 - 400 \sqrt{10} + 49 \sqrt{10} \cdot 9 + 70 \cdot 9}{2340}$

Step 10.11

Multiply $9$ by $49$ .

$\frac{- 2800 - 400 \sqrt{10} + 441 \sqrt{10} + 70 \cdot 9}{2340}$

Step 10.12

Multiply $70$ by $9$ .

$\frac{- 2800 - 400 \sqrt{10} + 441 \sqrt{10} + 630}{2340}$

Step 10.13

Add $- 2800$ and $630$ .

$\frac{- 2170 - 400 \sqrt{10} + 441 \sqrt{10}}{2340}$

Step 10.14

Add $- 400 \sqrt{10}$ and $441 \sqrt{10}$ .

$\frac{- 2170 + 41 \sqrt{10}}{2340}$

Step 11

Simplify with factoring out.

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Step 11.1

Rewrite $- 2170$ as $- 1 (2170)$ .

$\frac{- 1 (2170) + 41 \sqrt{10}}{2340}$

Step 11.2

Factor $- 1$ out of $41 \sqrt{10}$ .

$\frac{- 1 (2170) - (- 41 \sqrt{10})}{2340}$

Step 11.3

Factor $- 1$ out of $- 1 (2170) - (- 41 \sqrt{10})$ .

$\frac{- 1 (2170 - 41 \sqrt{10})}{2340}$

Step 11.4

Move the negative in front of the fraction.

$- \frac{2170 - 41 \sqrt{10}}{2340}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{2170 - 41 \sqrt{10}}{2340}$

Decimal Form:

$- 0.87194299 \dots$