Trigonometry Examples

${(- \frac{\sqrt{23}}{12})}^{2} - {(\frac{11}{12})}^{2}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Apply the product rule to $- \frac{\sqrt{23}}{12}$ .

${(- 1)}^{2} {(\frac{\sqrt{23}}{12})}^{2} - {(\frac{11}{12})}^{2}$

Step 1.1.2

Apply the product rule to $\frac{\sqrt{23}}{12}$ .

${(- 1)}^{2} \frac{{\sqrt{23}}^{2}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.2

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

$1 \frac{{\sqrt{23}}^{2}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.3

Multiply $\frac{{\sqrt{23}}^{2}}{12^{2}}$ by $1$ .

$\frac{{\sqrt{23}}^{2}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\frac{{(23^{\frac{1}{2}})}^{2}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4.2

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

$\frac{23^{\frac{1}{2} \cdot 2}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4.3

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

$\frac{23^{\frac{2}{2}}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4.4

Cancel the common factor of $2$ .

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Step 1.4.4.1

Cancel the common factor.

$\frac{23^{\frac{2}{2}}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4.4.2

Rewrite the expression.

$\frac{23^{1}}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.4.5

Evaluate the exponent.

$\frac{23}{12^{2}} - {(\frac{11}{12})}^{2}$

Step 1.5

Raise $12$ to the power of $2$ .

$\frac{23}{144} - {(\frac{11}{12})}^{2}$

Step 1.6

Apply the product rule to $\frac{11}{12}$ .

$\frac{23}{144} - \frac{11^{2}}{12^{2}}$

Step 1.7

Raise $11$ to the power of $2$ .

$\frac{23}{144} - \frac{121}{12^{2}}$

Step 1.8

Raise $12$ to the power of $2$ .

$\frac{23}{144} - \frac{121}{144}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{23 - 121}{144}$

Step 2.2

Subtract $121$ from $23$ .

$\frac{- 98}{144}$

Step 2.3

Cancel the common factor of $- 98$ and $144$ .

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Step 2.3.1

Factor $2$ out of $- 98$ .

$\frac{2 (- 49)}{144}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $144$ .

$\frac{2 \cdot - 49}{2 \cdot 72}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 49}{2 \cdot 72}$

Step 2.3.2.3

Rewrite the expression.

$\frac{- 49}{72}$

Step 2.4

Move the negative in front of the fraction.

$- \frac{49}{72}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{49}{72}$

Decimal Form:

$- 0.680\overline{5}$