Precalculus Examples

$\ln (\frac{e^{3} \sqrt[4]{x^{2}}}{\sqrt[3]{y}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$\ln (\frac{e^{3} \sqrt{\sqrt{x^{2}}}}{\sqrt[3]{y}})$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

$\ln (\frac{e^{3} \sqrt{x}}{\sqrt[3]{y}})$

Step 2

Multiply $\frac{e^{3} \sqrt{x}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\ln (\frac{e^{3} \sqrt{x}}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{e^{3} \sqrt{x}}{\sqrt[3]{y}}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}})$

Step 3.2

Raise $\sqrt[3]{y}$ to the power of $1$ .

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}})$

Step 3.3

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

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}})$

Step 3.4

Add $1$ and $2$ .

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}})$

Step 3.5

Rewrite ${\sqrt[3]{y}}^{3}$ as $y$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}})$

Step 3.5.2

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

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}})$

Step 3.5.3

Combine $\frac{1}{3}$ and $3$ .

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}})$

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}})$

Step 3.5.4.2

Rewrite the expression.

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{y^{1}})$

Step 3.5.5

Simplify.

$\ln (\frac{e^{3} \sqrt{x} {\sqrt[3]{y}}^{2}}{y})$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[3]{y}}^{2}$ as $\sqrt[3]{y^{2}}$ .

$\ln (\frac{e^{3} \sqrt{x} \sqrt[3]{y^{2}}}{y})$

Step 4.2

Combine exponents.

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Step 4.2.1

Rewrite the expression using the least common index of $6$ .

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Step 4.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

$\ln (\frac{e^{3} (y^{\frac{2}{3}} \sqrt{x})}{y})$

Step 4.2.1.2

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{4}{6}}$ .

$\ln (\frac{e^{3} (y^{\frac{4}{6}} \sqrt{x})}{y})$

Step 4.2.1.3

Rewrite $y^{\frac{4}{6}}$ as $\sqrt[6]{y^{4}}$ .

$\ln (\frac{e^{3} (\sqrt[6]{y^{4}} \sqrt{x})}{y})$

Step 4.2.1.4

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

$\ln (\frac{e^{3} (\sqrt[6]{y^{4}} x^{\frac{1}{2}})}{y})$

Step 4.2.1.5

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$\ln (\frac{e^{3} (\sqrt[6]{y^{4}} x^{\frac{3}{6}})}{y})$

Step 4.2.1.6

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$\ln (\frac{e^{3} (\sqrt[6]{y^{4}} \sqrt[6]{x^{3}})}{y})$

Step 4.2.2

Combine using the product rule for radicals.

$\ln (\frac{e^{3} \sqrt[6]{y^{4} x^{3}}}{y})$