Precalculus Examples

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

Step 1

Multiply $\frac{x^{4} {(x - 4)}^{2}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{x^{4} {(x - 4)}^{2}}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

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

Step 2.2

Raise $\sqrt{x^{2} + 1}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{x^{2} + 1}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

Rewrite ${\sqrt{x^{2} + 1}}^{2}$ as $x^{2} + 1$ .

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Simplify.

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

Step 3

Reorder factors in $\ln (\frac{x^{4} {(x - 4)}^{2} \sqrt{x^{2} + 1}}{x^{2} + 1})$ .

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