Trigonometry Examples

${(\frac{2}{\sqrt{13}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

${(\frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

${(\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.2

Raise $\sqrt{13}$ to the power of $1$ .

${(\frac{2 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.3

Raise $\sqrt{13}$ to the power of $1$ .

${(\frac{2 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.4

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

${(\frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.5

Add $1$ and $1$ .

${(\frac{2 \sqrt{13}}{{\sqrt{13}}^{2}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

${(\frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.6.2

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

${(\frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.6.3

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

${(\frac{2 \sqrt{13}}{13^{\frac{2}{2}}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

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.

${(\frac{2 \sqrt{13}}{13^{\frac{2}{2}}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.6.4.2

Rewrite the expression.

${(\frac{2 \sqrt{13}}{13^{1}})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.2.6.5

Evaluate the exponent.

${(\frac{2 \sqrt{13}}{13})}^{2} - {(\frac{3}{\sqrt{13}})}^{2}$

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 $\frac{2 \sqrt{13}}{13}$ .

$\frac{{(2 \sqrt{13})}^{2}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.3.2

Apply the product rule to $2 \sqrt{13}$ .

$\frac{2^{2} {\sqrt{13}}^{2}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

$\frac{4 {\sqrt{13}}^{2}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{4 {(13^{\frac{1}{2}})}^{2}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2.2

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

$\frac{4 \cdot 13^{\frac{1}{2} \cdot 2}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2.3

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

$\frac{4 \cdot 13^{\frac{2}{2}}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2.4

Cancel the common factor of $2$ .

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Step 1.4.2.4.1

Cancel the common factor.

$\frac{4 \cdot 13^{\frac{2}{2}}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2.4.2

Rewrite the expression.

$\frac{4 \cdot 13^{1}}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.4.2.5

Evaluate the exponent.

$\frac{4 \cdot 13}{13^{2}} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.5

Raise $13$ to the power of $2$ .

$\frac{4 \cdot 13}{169} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.6

Multiply $4$ by $13$ .

$\frac{52}{169} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.7

Cancel the common factor of $52$ and $169$ .

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Step 1.7.1

Factor $13$ out of $52$ .

$\frac{13 (4)}{169} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $13$ out of $169$ .

$\frac{13 \cdot 4}{13 \cdot 13} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.7.2.2

Cancel the common factor.

$\frac{13 \cdot 4}{13 \cdot 13} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.7.2.3

Rewrite the expression.

$\frac{4}{13} - {(\frac{3}{\sqrt{13}})}^{2}$

Step 1.8

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{4}{13} - {(\frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})}^{2}$

Step 1.9

Combine and simplify the denominator.

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Step 1.9.1

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}})}^{2}$

Step 1.9.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}})}^{2}$

Step 1.9.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}})}^{2}$

Step 1.9.4

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

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})}^{2}$

Step 1.9.5

Add $1$ and $1$ .

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{{\sqrt{13}}^{2}})}^{2}$

Step 1.9.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})}^{2}$

Step 1.9.6.2

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

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})}^{2}$

Step 1.9.6.3

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

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{13^{\frac{2}{2}}})}^{2}$

Step 1.9.6.4

Cancel the common factor of $2$ .

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Step 1.9.6.4.1

Cancel the common factor.

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{13^{\frac{2}{2}}})}^{2}$

Step 1.9.6.4.2

Rewrite the expression.

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{13^{1}})}^{2}$

Step 1.9.6.5

Evaluate the exponent.

$\frac{4}{13} - {(\frac{3 \sqrt{13}}{13})}^{2}$

Step 1.10

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

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Step 1.10.1

Apply the product rule to $\frac{3 \sqrt{13}}{13}$ .

$\frac{4}{13} - \frac{{(3 \sqrt{13})}^{2}}{13^{2}}$

Step 1.10.2

Apply the product rule to $3 \sqrt{13}$ .

$\frac{4}{13} - \frac{3^{2} {\sqrt{13}}^{2}}{13^{2}}$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Raise $3$ to the power of $2$ .

$\frac{4}{13} - \frac{9 {\sqrt{13}}^{2}}{13^{2}}$

Step 1.11.2

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.11.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{4}{13} - \frac{9 {(13^{\frac{1}{2}})}^{2}}{13^{2}}$

Step 1.11.2.2

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

$\frac{4}{13} - \frac{9 \cdot 13^{\frac{1}{2} \cdot 2}}{13^{2}}$

Step 1.11.2.3

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

$\frac{4}{13} - \frac{9 \cdot 13^{\frac{2}{2}}}{13^{2}}$

Step 1.11.2.4

Cancel the common factor of $2$ .

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Step 1.11.2.4.1

Cancel the common factor.

$\frac{4}{13} - \frac{9 \cdot 13^{\frac{2}{2}}}{13^{2}}$

Step 1.11.2.4.2

Rewrite the expression.

$\frac{4}{13} - \frac{9 \cdot 13^{1}}{13^{2}}$

Step 1.11.2.5

Evaluate the exponent.

$\frac{4}{13} - \frac{9 \cdot 13}{13^{2}}$

Step 1.12

Raise $13$ to the power of $2$ .

$\frac{4}{13} - \frac{9 \cdot 13}{169}$

Step 1.13

Multiply $9$ by $13$ .

$\frac{4}{13} - \frac{117}{169}$

Step 1.14

Cancel the common factor of $117$ and $169$ .

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Step 1.14.1

Factor $13$ out of $117$ .

$\frac{4}{13} - \frac{13 (9)}{169}$

Step 1.14.2

Cancel the common factors.

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Step 1.14.2.1

Factor $13$ out of $169$ .

$\frac{4}{13} - \frac{13 \cdot 9}{13 \cdot 13}$

Step 1.14.2.2

Cancel the common factor.

$\frac{4}{13} - \frac{13 \cdot 9}{13 \cdot 13}$

Step 1.14.2.3

Rewrite the expression.

$\frac{4}{13} - \frac{9}{13}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{4 - 9}{13}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Subtract $9$ from $4$ .

$\frac{- 5}{13}$

Step 2.2.2

Move the negative in front of the fraction.

$- \frac{5}{13}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{13}$

Decimal Form:

$- 0.\overline{384615}$