Trigonometry Examples

$\frac{42 - 20 \sqrt{21} (21 \sqrt{21} - 40)}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the distributive property.

$\frac{42 - 20 \sqrt{21} (21 \sqrt{21}) - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.2

Multiply $- 20 \sqrt{21} (21 \sqrt{21})$ .

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Step 1.2.1

Multiply $21$ by $- 20$ .

$\frac{42 - 420 \sqrt{21} \sqrt{21} - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.2.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{42 - 420 ({\sqrt{21}}^{1} \sqrt{21}) - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.2.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{42 - 420 ({\sqrt{21}}^{1} {\sqrt{21}}^{1}) - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.2.4

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

$\frac{42 - 420 {\sqrt{21}}^{1 + 1} - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{42 - 420 {\sqrt{21}}^{2} - 20 \sqrt{21} \cdot - 40}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.3

Multiply $- 40$ by $- 20$ .

$\frac{42 - 420 {\sqrt{21}}^{2} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4

Simplify each term.

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Step 1.4.1

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{42 - 420 {(21^{\frac{1}{2}})}^{2} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.1.2

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

$\frac{42 - 420 \cdot 21^{\frac{1}{2} \cdot 2} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.1.3

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

$\frac{42 - 420 \cdot 21^{\frac{2}{2}} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.1.4

Cancel the common factor of $2$ .

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Step 1.4.1.4.1

Cancel the common factor.

$\frac{42 - 420 \cdot 21^{\frac{2}{2}} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.1.4.2

Rewrite the expression.

$\frac{42 - 420 \cdot 21^{1} + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.1.5

Evaluate the exponent.

$\frac{42 - 420 \cdot 21 + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.4.2

Multiply $- 420$ by $21$ .

$\frac{42 - 8820 + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 1.5

Subtract $8820$ from $42$ .

$\frac{- 8778 + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21} - 40)}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the distributive property.

$\frac{- 8778 + 800 \sqrt{21}}{21 \sqrt{21} + 40 (21 \sqrt{21}) + 40 \cdot - 40}$

Step 2.2

Multiply $21$ by $40$ .

$\frac{- 8778 + 800 \sqrt{21}}{21 \sqrt{21} + 840 \sqrt{21} + 40 \cdot - 40}$

Step 2.3

Multiply $40$ by $- 40$ .

$\frac{- 8778 + 800 \sqrt{21}}{21 \sqrt{21} + 840 \sqrt{21} - 1600}$

Step 2.4

Add $21 \sqrt{21}$ and $840 \sqrt{21}$ .

$\frac{- 8778 + 800 \sqrt{21}}{861 \sqrt{21} - 1600}$

Step 3

Multiply $\frac{- 8778 + 800 \sqrt{21}}{861 \sqrt{21} - 1600}$ by $\frac{861 \sqrt{21} + 1600}{861 \sqrt{21} + 1600}$ .

$\frac{- 8778 + 800 \sqrt{21}}{861 \sqrt{21} - 1600} \cdot \frac{861 \sqrt{21} + 1600}{861 \sqrt{21} + 1600}$

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{- 8778 + 800 \sqrt{21}}{861 \sqrt{21} - 1600}$ by $\frac{861 \sqrt{21} + 1600}{861 \sqrt{21} + 1600}$ .

$\frac{(- 8778 + 800 \sqrt{21}) (861 \sqrt{21} + 1600)}{(861 \sqrt{21} - 1600) (861 \sqrt{21} + 1600)}$

Step 4.2

Expand the denominator using the FOIL method.

$\frac{(- 8778 + 800 \sqrt{21}) (861 \sqrt{21} + 1600)}{741321 {\sqrt{21}}^{2} + 1377600 \sqrt{21} - 1377600 \sqrt{21} - 2560000}$

Step 4.3

Simplify.

$\frac{(- 8778 + 800 \sqrt{21}) (861 \sqrt{21} + 1600)}{13007741}$

Step 5

Expand $(- 8778 + 800 \sqrt{21}) (861 \sqrt{21} + 1600)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{- 8778 (861 \sqrt{21} + 1600) + 800 \sqrt{21} (861 \sqrt{21} + 1600)}{13007741}$

Step 5.2

Apply the distributive property.

$\frac{- 8778 (861 \sqrt{21}) - 8778 \cdot 1600 + 800 \sqrt{21} (861 \sqrt{21} + 1600)}{13007741}$

Step 5.3

Apply the distributive property.

$\frac{- 8778 (861 \sqrt{21}) - 8778 \cdot 1600 + 800 \sqrt{21} (861 \sqrt{21}) + 800 \sqrt{21} \cdot 1600}{13007741}$

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 $861$ by $- 8778$ .

$\frac{- 7557858 \sqrt{21} - 8778 \cdot 1600 + 800 \sqrt{21} (861 \sqrt{21}) + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.2

Multiply $- 8778$ by $1600$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 800 \sqrt{21} (861 \sqrt{21}) + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.3

Multiply $800 \sqrt{21} (861 \sqrt{21})$ .

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Step 6.1.3.1

Multiply $861$ by $800$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \sqrt{21} \sqrt{21} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.3.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 ({\sqrt{21}}^{1} \sqrt{21}) + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.3.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 ({\sqrt{21}}^{1} {\sqrt{21}}^{1}) + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.3.4

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

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 {\sqrt{21}}^{1 + 1} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.3.5

Add $1$ and $1$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 {\sqrt{21}}^{2} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 6.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 {(21^{\frac{1}{2}})}^{2} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4.2

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

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \cdot 21^{\frac{1}{2} \cdot 2} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4.3

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

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \cdot 21^{\frac{2}{2}} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4.4

Cancel the common factor of $2$ .

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Step 6.1.4.4.1

Cancel the common factor.

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \cdot 21^{\frac{2}{2}} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4.4.2

Rewrite the expression.

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \cdot 21^{1} + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.4.5

Evaluate the exponent.

$\frac{- 7557858 \sqrt{21} - 14044800 + 688800 \cdot 21 + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.5

Multiply $688800$ by $21$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 14464800 + 800 \sqrt{21} \cdot 1600}{13007741}$

Step 6.1.6

Multiply $1600$ by $800$ .

$\frac{- 7557858 \sqrt{21} - 14044800 + 14464800 + 1280000 \sqrt{21}}{13007741}$

Step 6.2

Add $- 7557858 \sqrt{21}$ and $1280000 \sqrt{21}$ .

$\frac{- 14044800 + 14464800 - 6277858 \sqrt{21}}{13007741}$

Step 6.3

Add $- 14044800$ and $14464800$ .

$\frac{420000 - 6277858 \sqrt{21}}{13007741}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{420000 - 6277858 \sqrt{21}}{13007741}$

Decimal Form:

$- 2.17937607 \dots$