Calculus Examples

$f^{3} (x) = 12 x + 16$

Step 1

Divide each term in $f^{3} x = 12 x + 16$ by $x$ and simplify.

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Step 1.1

Divide each term in $f^{3} x = 12 x + 16$ by $x$ .

$\frac{f^{3} x}{x} = \frac{12 x}{x} + \frac{16}{x}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{f^{3} x}{x} = \frac{12 x}{x} + \frac{16}{x}$

Step 1.2.1.2

Divide $f^{3}$ by $1$ .

$f^{3} = \frac{12 x}{x} + \frac{16}{x}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $x$ .

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Step 1.3.1.1

Cancel the common factor.

$f^{3} = \frac{12 x}{x} + \frac{16}{x}$

Step 1.3.1.2

Divide $12$ by $1$ .

$f^{3} = 12 + \frac{16}{x}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$f = \sqrt[3]{12 + \frac{16}{x}}$

Step 3

Simplify $\sqrt[3]{12 + \frac{16}{x}}$ .

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Step 3.1

Factor $4$ out of $12 + \frac{16}{x}$ .

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Step 3.1.1

Factor $4$ out of $12$ .

$f = \sqrt[3]{4 (3) + \frac{16}{x}}$

Step 3.1.2

Factor $4$ out of $\frac{16}{x}$ .

$f = \sqrt[3]{4 \cdot 3 + 4 \frac{4}{x}}$

Step 3.1.3

Factor $4$ out of $4 \cdot 3 + 4 \frac{4}{x}$ .

$f = \sqrt[3]{4 (3 + \frac{4}{x})}$

Step 3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f = \sqrt[3]{4 (\frac{3 x}{x} + \frac{4}{x})}$

Step 3.3

Combine the numerators over the common denominator.

$f = \sqrt[3]{4 \frac{3 x + 4}{x}}$

Step 3.4

Combine $4$ and $\frac{3 x + 4}{x}$ .

$f = \sqrt[3]{\frac{4 (3 x + 4)}{x}}$

Step 3.5

Rewrite $\sqrt[3]{\frac{4 (3 x + 4)}{x}}$ as $\frac{\sqrt[3]{4 (3 x + 4)}}{\sqrt[3]{x}}$ .

$f = \frac{\sqrt[3]{4 (3 x + 4)}}{\sqrt[3]{x}}$

Step 3.6

Multiply $\frac{\sqrt[3]{4 (3 x + 4)}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$f = \frac{\sqrt[3]{4 (3 x + 4)}}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Step 3.7

Combine and simplify the denominator.

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Step 3.7.1

Multiply $\frac{\sqrt[3]{4 (3 x + 4)}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 3.7.2

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Step 3.7.3

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 3.7.4

Add $1$ and $2$ .

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

Step 3.7.5

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

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Step 3.7.5.1

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

Step 3.7.5.2

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 3.7.5.3

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.7.5.4

Cancel the common factor of $3$ .

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Step 3.7.5.4.1

Cancel the common factor.

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.7.5.4.2

Rewrite the expression.

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{x^{1}}$

Step 3.7.5.5

Simplify.

$f = \frac{\sqrt[3]{4 (3 x + 4)} {\sqrt[3]{x}}^{2}}{x}$

Step 3.8

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

$f = \frac{\sqrt[3]{4 (3 x + 4)} \sqrt[3]{x^{2}}}{x}$

Step 3.9

Combine using the product rule for radicals.

$f = \frac{\sqrt[3]{4 (3 x + 4) x^{2}}}{x}$

Step 3.10

Reorder factors in $\frac{\sqrt[3]{4 (3 x + 4) x^{2}}}{x}$ .

$f = \frac{\sqrt[3]{4 x^{2} (3 x + 4)}}{x}$

Step 4

To rewrite as a function of $x$ , write the equation so that $f$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{\sqrt[3]{4 x^{2} (3 x + 4)}}{x}$