Precalculus Examples

$\ln (\frac{x^{\frac{1}{3}} \sqrt{x^{4} - x}}{e^{2 x}})$

Step 1

Rewrite $\frac{x^{\frac{1}{3}} \sqrt{x^{4} - x}}{e^{2 x}}$ as $x^{\frac{1}{3}} \sqrt{x^{4} - x} {(e^{2 x})}^{- 1}$ .

$\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x} {(e^{2 x})}^{- 1})$

Step 2

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

$\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x} e^{2 x \cdot - 1})$

Step 3

Rewrite $\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x} e^{2 x \cdot - 1})$ as $\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x}) + \ln (e^{2 x \cdot - 1})$ .

$\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x}) + \ln (e^{2 x \cdot - 1})$

Step 4

Use logarithm rules to move $2 x \cdot - 1$ out of the exponent.

$\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x}) + 2 x \cdot - 1 \ln (e)$

Step 5

Multiply $- 1$ by $2$ .

$\ln (x^{\frac{1}{3}} \sqrt{x^{4} - x}) - 2 x \ln (e)$

Step 6

The natural logarithm of $e$ is $1$ .

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

Step 7

Multiply $- 2$ by $1$ .

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

Step 8

Simplify each term.

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Step 8.1

Factor $x$ out of $x^{4} - x$ .

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Step 8.1.1

Factor $x$ out of $x^{4}$ .

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

Step 8.1.2

Factor $x$ out of $- x$ .

$\ln (x^{\frac{1}{3}} \sqrt{x \cdot x^{3} + x \cdot - 1}) - 2 x$

Step 8.1.3

Factor $x$ out of $x \cdot x^{3} + x \cdot - 1$ .

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

Step 8.2

Rewrite $1$ as $1^{3}$ .

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

Step 8.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$\ln (x^{\frac{1}{3}} \sqrt{x ((x - 1) (x^{2} + x \cdot 1 + 1^{2}))}) - 2 x$

Step 8.4

Simplify.

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Step 8.4.1

Multiply $x$ by $1$ .

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

Step 8.4.2

One to any power is one.

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

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

Step 9

Reorder $\ln (x^{\frac{1}{3}} \sqrt{x (x - 1) (x^{2} + x + 1)})$ and $- 2 x$ .

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