Calculus Examples

$- \frac{11}{12 \sqrt{12^{2} - 121}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$- \frac{11}{12 \sqrt{144 - 121}}$

Step 1.2

Subtract $121$ from $144$ .

$- \frac{11}{12 \sqrt{23}}$

Step 2

Multiply $\frac{11}{12 \sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{11}{12 \sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$- \frac{11 \sqrt{23}}{12 \sqrt{23} \sqrt{23}}$

Step 3.2

Move $\sqrt{23}$ .

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

Step 3.3

Raise $\sqrt{23}$ to the power of $1$ .

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

Step 3.4

Raise $\sqrt{23}$ to the power of $1$ .

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

Step 3.5

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

$- \frac{11 \sqrt{23}}{12 {\sqrt{23}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

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

Step 3.7.4.2

Rewrite the expression.

$- \frac{11 \sqrt{23}}{12 \cdot 23^{1}}$

Step 3.7.5

Evaluate the exponent.

$- \frac{11 \sqrt{23}}{12 \cdot 23}$

Step 4

Multiply $12$ by $23$ .

$- \frac{11 \sqrt{23}}{276}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{11 \sqrt{23}}{276}$

Decimal Form:

$- 0.19113821 \dots$