Calculus Examples

$\frac{- 8}{\sqrt[12]{- 8 x^{2} - 9 x - 2}}$

Step 1

Move the negative in front of the fraction.

$- \frac{8}{\sqrt[12]{- 8 x^{2} - 9 x - 2}}$

Step 2

Multiply $\frac{8}{\sqrt[12]{- 8 x^{2} - 9 x - 2}}$ by $\frac{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}$ .

$- (\frac{8}{\sqrt[12]{- 8 x^{2} - 9 x - 2}} \cdot \frac{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{8}{\sqrt[12]{- 8 x^{2} - 9 x - 2}}$ by $\frac{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}$ .

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{\sqrt[12]{- 8 x^{2} - 9 x - 2} {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}$

Step 3.2

Raise $\sqrt[12]{- 8 x^{2} - 9 x - 2}$ to the power of $1$ .

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{1} {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}$

Step 3.3

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

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{1 + 11}}$

Step 3.4

Add $1$ and $11$ .

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{12}}$

Step 3.5

Rewrite ${\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{12}$ as $- 8 x^{2} - 9 x - 2$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{- 8 x^{2} - 9 x - 2}$ as ${(- 8 x^{2} - 9 x - 2)}^{\frac{1}{12}}$ .

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{({(- 8 x^{2} - 9 x - 2)}^{\frac{1}{12}})}^{12}}$

Step 3.5.2

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

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{(- 8 x^{2} - 9 x - 2)}^{\frac{1}{12} \cdot 12}}$

Step 3.5.3

Combine $\frac{1}{12}$ and $12$ .

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{(- 8 x^{2} - 9 x - 2)}^{\frac{12}{12}}}$

Step 3.5.4

Cancel the common factor of $12$ .

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Step 3.5.4.1

Cancel the common factor.

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{(- 8 x^{2} - 9 x - 2)}^{\frac{12}{12}}}$

Step 3.5.4.2

Rewrite the expression.

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{{(- 8 x^{2} - 9 x - 2)}^{1}}$

Step 3.5.5

Simplify.

$- \frac{8 {\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}}{- 8 x^{2} - 9 x - 2}$

Step 4

Rewrite ${\sqrt[12]{- 8 x^{2} - 9 x - 2}}^{11}$ as $\sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- 8 x^{2} - 9 x - 2}$

Step 5

Factor $- 1$ out of $- 8 x^{2}$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- (8 x^{2}) - 9 x - 2}$

Step 6

Factor $- 1$ out of $- 9 x$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- (8 x^{2}) - (9 x) - 2}$

Step 7

Factor $- 1$ out of $- (8 x^{2}) - (9 x)$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- (8 x^{2} + 9 x) - 2}$

Step 8

Rewrite $- 2$ as $- 1 (2)$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- (8 x^{2} + 9 x) - 1 (2)}$

Step 9

Factor $- 1$ out of $- (8 x^{2} + 9 x) - 1 (2)$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- (8 x^{2} + 9 x + 2)}$

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- (8 x^{2} + 9 x + 2)$ as $- 1 (8 x^{2} + 9 x + 2)$ .

$- \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{- 1 (8 x^{2} + 9 x + 2)}$

Step 10.2

Move the negative in front of the fraction.

$- - \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{8 x^{2} + 9 x + 2}$

Step 10.3

Multiply $- 1$ by $- 1$ .

$1 \frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{8 x^{2} + 9 x + 2}$

Step 10.4

Multiply $\frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{8 x^{2} + 9 x + 2}$ by $1$ .

$\frac{8 \sqrt[12]{{(- 8 x^{2} - 9 x - 2)}^{11}}}{8 x^{2} + 9 x + 2}$