Trigonometry Examples

${(- \frac{2}{\sqrt{29}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

${(- (\frac{2}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}}))}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

${(- \frac{2 \sqrt{29}}{\sqrt{29} \sqrt{29}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.2

Raise $\sqrt{29}$ to the power of $1$ .

${(- \frac{2 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.3

Raise $\sqrt{29}$ to the power of $1$ .

${(- \frac{2 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.4

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

${(- \frac{2 \sqrt{29}}{{\sqrt{29}}^{1 + 1}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.5

Add $1$ and $1$ .

${(- \frac{2 \sqrt{29}}{{\sqrt{29}}^{2}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

${(- \frac{2 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6.2

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

${(- \frac{2 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6.3

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

${(- \frac{2 \sqrt{29}}{29^{\frac{2}{2}}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

${(- \frac{2 \sqrt{29}}{29^{\frac{2}{2}}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6.4.2

Rewrite the expression.

${(- \frac{2 \sqrt{29}}{29^{1}})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.2.6.5

Evaluate the exponent.

${(- \frac{2 \sqrt{29}}{29})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.3.1

Apply the product rule to $- \frac{2 \sqrt{29}}{29}$ .

${(- 1)}^{2} {(\frac{2 \sqrt{29}}{29})}^{2} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.3.2

Apply the product rule to $\frac{2 \sqrt{29}}{29}$ .

${(- 1)}^{2} \frac{{(2 \sqrt{29})}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.3.3

Apply the product rule to $2 \sqrt{29}$ .

${(- 1)}^{2} \frac{2^{2} {\sqrt{29}}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.4

Raise $- 1$ to the power of $2$ .

$1 \frac{2^{2} {\sqrt{29}}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.5

Multiply $\frac{2^{2} {\sqrt{29}}^{2}}{29^{2}}$ by $1$ .

$\frac{2^{2} {\sqrt{29}}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Raise $2$ to the power of $2$ .

$\frac{4 {\sqrt{29}}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 1.6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\frac{4 {(29^{\frac{1}{2}})}^{2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2.2

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

$\frac{4 \cdot 29^{\frac{1}{2} \cdot 2}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2.3

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

$\frac{4 \cdot 29^{\frac{2}{2}}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2.4

Cancel the common factor of $2$ .

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Step 1.6.2.4.1

Cancel the common factor.

$\frac{4 \cdot 29^{\frac{2}{2}}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2.4.2

Rewrite the expression.

$\frac{4 \cdot 29^{1}}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.6.2.5

Evaluate the exponent.

$\frac{4 \cdot 29}{29^{2}} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.7

Raise $29$ to the power of $2$ .

$\frac{4 \cdot 29}{841} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.8

Multiply $4$ by $29$ .

$\frac{116}{841} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.9

Cancel the common factor of $116$ and $841$ .

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Step 1.9.1

Factor $29$ out of $116$ .

$\frac{29 (4)}{841} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $29$ out of $841$ .

$\frac{29 \cdot 4}{29 \cdot 29} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.9.2.2

Cancel the common factor.

$\frac{29 \cdot 4}{29 \cdot 29} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.9.2.3

Rewrite the expression.

$\frac{4}{29} - {(- \frac{5}{\sqrt{29}})}^{2}$

Step 1.10

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\frac{4}{29} - {(- (\frac{5}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}}))}^{2}$

Step 1.11

Combine and simplify the denominator.

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Step 1.11.1

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{\sqrt{29} \sqrt{29}})}^{2}$

Step 1.11.2

Raise $\sqrt{29}$ to the power of $1$ .

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}})}^{2}$

Step 1.11.3

Raise $\sqrt{29}$ to the power of $1$ .

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}})}^{2}$

Step 1.11.4

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

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{{\sqrt{29}}^{1 + 1}})}^{2}$

Step 1.11.5

Add $1$ and $1$ .

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{{\sqrt{29}}^{2}})}^{2}$

Step 1.11.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 1.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}})}^{2}$

Step 1.11.6.2

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

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}})}^{2}$

Step 1.11.6.3

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

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{29^{\frac{2}{2}}})}^{2}$

Step 1.11.6.4

Cancel the common factor of $2$ .

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Step 1.11.6.4.1

Cancel the common factor.

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{29^{\frac{2}{2}}})}^{2}$

Step 1.11.6.4.2

Rewrite the expression.

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{29^{1}})}^{2}$

Step 1.11.6.5

Evaluate the exponent.

$\frac{4}{29} - {(- \frac{5 \sqrt{29}}{29})}^{2}$

Step 1.12

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.12.1

Apply the product rule to $- \frac{5 \sqrt{29}}{29}$ .

$\frac{4}{29} - ({(- 1)}^{2} {(\frac{5 \sqrt{29}}{29})}^{2})$

Step 1.12.2

Apply the product rule to $\frac{5 \sqrt{29}}{29}$ .

$\frac{4}{29} - ({(- 1)}^{2} \frac{{(5 \sqrt{29})}^{2}}{29^{2}})$

Step 1.12.3

Apply the product rule to $5 \sqrt{29}$ .

$\frac{4}{29} - ({(- 1)}^{2} \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}})$

Step 1.13

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.13.1

Move ${(- 1)}^{2}$ .

$\frac{4}{29} + {(- 1)}^{2} \cdot - 1 \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}}$

Step 1.13.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.13.2.1

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

$\frac{4}{29} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}}$

Step 1.13.2.2

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

$\frac{4}{29} + {(- 1)}^{2 + 1} \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}}$

Step 1.13.3

Add $2$ and $1$ .

$\frac{4}{29} + {(- 1)}^{3} \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}}$

Step 1.14

Raise $- 1$ to the power of $3$ .

$\frac{4}{29} - \frac{5^{2} {\sqrt{29}}^{2}}{29^{2}}$

Step 1.15

Simplify the numerator.

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Step 1.15.1

Raise $5$ to the power of $2$ .

$\frac{4}{29} - \frac{25 {\sqrt{29}}^{2}}{29^{2}}$

Step 1.15.2

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 1.15.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\frac{4}{29} - \frac{25 {(29^{\frac{1}{2}})}^{2}}{29^{2}}$

Step 1.15.2.2

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

$\frac{4}{29} - \frac{25 \cdot 29^{\frac{1}{2} \cdot 2}}{29^{2}}$

Step 1.15.2.3

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

$\frac{4}{29} - \frac{25 \cdot 29^{\frac{2}{2}}}{29^{2}}$

Step 1.15.2.4

Cancel the common factor of $2$ .

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Step 1.15.2.4.1

Cancel the common factor.

$\frac{4}{29} - \frac{25 \cdot 29^{\frac{2}{2}}}{29^{2}}$

Step 1.15.2.4.2

Rewrite the expression.

$\frac{4}{29} - \frac{25 \cdot 29^{1}}{29^{2}}$

Step 1.15.2.5

Evaluate the exponent.

$\frac{4}{29} - \frac{25 \cdot 29}{29^{2}}$

Step 1.16

Raise $29$ to the power of $2$ .

$\frac{4}{29} - \frac{25 \cdot 29}{841}$

Step 1.17

Multiply $25$ by $29$ .

$\frac{4}{29} - \frac{725}{841}$

Step 1.18

Cancel the common factor of $725$ and $841$ .

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Step 1.18.1

Factor $29$ out of $725$ .

$\frac{4}{29} - \frac{29 (25)}{841}$

Step 1.18.2

Cancel the common factors.

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Step 1.18.2.1

Factor $29$ out of $841$ .

$\frac{4}{29} - \frac{29 \cdot 25}{29 \cdot 29}$

Step 1.18.2.2

Cancel the common factor.

$\frac{4}{29} - \frac{29 \cdot 25}{29 \cdot 29}$

Step 1.18.2.3

Rewrite the expression.

$\frac{4}{29} - \frac{25}{29}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{4 - 25}{29}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Subtract $25$ from $4$ .

$\frac{- 21}{29}$

Step 2.2.2

Move the negative in front of the fraction.

$- \frac{21}{29}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{21}{29}$

Decimal Form:

$- 0.72413793 \dots$