Calculus Examples

$f x = \sqrt[3]{x}$

Step 1

Rewrite the equation as $\sqrt[3]{x} = f x$ .

$\sqrt[3]{x} = f x$

Step 2

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x}}^{3} = {(f x)}^{3}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

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

${(x^{\frac{1}{3}})}^{3} = {(f x)}^{3}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.2.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = {(f x)}^{3}$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = {(f x)}^{3}$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(f x)}^{3}$

Step 3.2.1.2

Simplify.

$x = {(f x)}^{3}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Apply the product rule to $f x$ .

$x = f^{3} x^{3}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Subtract $f^{3} x^{3}$ from both sides of the equation.

$x - f^{3} x^{3} = 0$

Step 4.2

Factor $x$ out of $x - f^{3} x^{3}$ .

Tap for more steps...

Step 4.2.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - f^{3} x^{3} = 0$

Step 4.2.2

Factor $x$ out of $- f^{3} x^{3}$ .

$x \cdot 1 + x (- f^{3} x^{2}) = 0$

Step 4.2.3

Factor $x$ out of $x \cdot 1 + x (- f^{3} x^{2})$ .

$x (1 - f^{3} x^{2}) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - f^{3} x^{2} = 0$

Step 4.4

Set $x$ equal to $0$ .

$x = 0$

Step 4.5

Set $1 - f^{3} x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $1 - f^{3} x^{2}$ equal to $0$ .

$1 - f^{3} x^{2} = 0$

Step 4.5.2

Solve $1 - f^{3} x^{2} = 0$ for $x$ .

Tap for more steps...

Step 4.5.2.1

Subtract $1$ from both sides of the equation.

$- f^{3} x^{2} = - 1$

Step 4.5.2.2

Divide each term in $- f^{3} x^{2} = - 1$ by $- f^{3}$ and simplify.

Tap for more steps...

Step 4.5.2.2.1

Divide each term in $- f^{3} x^{2} = - 1$ by $- f^{3}$ .

$\frac{- f^{3} x^{2}}{- f^{3}} = \frac{- 1}{- f^{3}}$

Step 4.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{f^{3} x^{2}}{f^{3}} = \frac{- 1}{- f^{3}}$

Step 4.5.2.2.2.2

Cancel the common factor of $f^{3}$ .

Tap for more steps...

Step 4.5.2.2.2.2.1

Cancel the common factor.

$\frac{f^{3} x^{2}}{f^{3}} = \frac{- 1}{- f^{3}}$

Step 4.5.2.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{- f^{3}}$

Step 4.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.5.2.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{1}{f^{3}}$

Step 4.5.2.3

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

$x = \pm \sqrt{\frac{1}{f^{3}}}$

Step 4.5.2.4

Simplify $\pm \sqrt{\frac{1}{f^{3}}}$ .

Tap for more steps...

Step 4.5.2.4.1

Rewrite $\frac{1}{f^{3}}$ as ${(\frac{1}{f})}^{2} \frac{1}{f}$ .

Tap for more steps...

Step 4.5.2.4.1.1

Factor the perfect power $1^{2}$ out of $1$ .

$x = \pm \sqrt{\frac{1^{2} \cdot 1}{f^{3}}}$

Step 4.5.2.4.1.2

Factor the perfect power $f^{2}$ out of $f^{3}$ .

$x = \pm \sqrt{\frac{1^{2} \cdot 1}{f^{2} f}}$

Step 4.5.2.4.1.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{f^{2} f}$ .

$x = \pm \sqrt{{(\frac{1}{f})}^{2} \frac{1}{f}}$

Step 4.5.2.4.2

Pull terms out from under the radical.

$x = \pm \frac{1}{f} \sqrt{\frac{1}{f}}$

Step 4.5.2.4.3

Rewrite $\sqrt{\frac{1}{f}}$ as $\frac{\sqrt{1}}{\sqrt{f}}$ .

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{1}}{\sqrt{f}}$

Step 4.5.2.4.4

Any root of $1$ is $1$ .

$x = \pm \frac{1}{f} \cdot \frac{1}{\sqrt{f}}$

Step 4.5.2.4.5

Multiply $\frac{1}{\sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$x = \pm \frac{1}{f} (\frac{1}{\sqrt{f}} \cdot \frac{\sqrt{f}}{\sqrt{f}})$

Step 4.5.2.4.6

Combine and simplify the denominator.

Tap for more steps...

Step 4.5.2.4.6.1

Multiply $\frac{1}{\sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{\sqrt{f} \sqrt{f}}$

Step 4.5.2.4.6.2

Raise $\sqrt{f}$ to the power of $1$ .

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{{\sqrt{f}}^{1} \sqrt{f}}$

Step 4.5.2.4.6.3

Raise $\sqrt{f}$ to the power of $1$ .

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{{\sqrt{f}}^{1} {\sqrt{f}}^{1}}$

Step 4.5.2.4.6.4

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

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{{\sqrt{f}}^{1 + 1}}$

Step 4.5.2.4.6.5

Add $1$ and $1$ .

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{{\sqrt{f}}^{2}}$

Step 4.5.2.4.6.6

Rewrite ${\sqrt{f}}^{2}$ as $f$ .

Tap for more steps...

Step 4.5.2.4.6.6.1

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

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{{(f^{\frac{1}{2}})}^{2}}$

Step 4.5.2.4.6.6.2

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

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{f^{\frac{1}{2} \cdot 2}}$

Step 4.5.2.4.6.6.3

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

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{f^{\frac{2}{2}}}$

Step 4.5.2.4.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.2.4.6.6.4.1

Cancel the common factor.

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{f^{\frac{2}{2}}}$

Step 4.5.2.4.6.6.4.2

Rewrite the expression.

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{f^{1}}$

Step 4.5.2.4.6.6.5

Simplify.

$x = \pm \frac{1}{f} \cdot \frac{\sqrt{f}}{f}$

Step 4.5.2.4.7

Multiply $\frac{1}{f} \cdot \frac{\sqrt{f}}{f}$ .

Tap for more steps...

Step 4.5.2.4.7.1

Multiply $\frac{1}{f}$ by $\frac{\sqrt{f}}{f}$ .

$x = \pm \frac{\sqrt{f}}{f \cdot f}$

Step 4.5.2.4.7.2

Raise $f$ to the power of $1$ .

$x = \pm \frac{\sqrt{f}}{f^{1} f}$

Step 4.5.2.4.7.3

Raise $f$ to the power of $1$ .

$x = \pm \frac{\sqrt{f}}{f^{1} f^{1}}$

Step 4.5.2.4.7.4

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

$x = \pm \frac{\sqrt{f}}{f^{1 + 1}}$

Step 4.5.2.4.7.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{f}}{f^{2}}$

Step 4.5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{f}}{f^{2}}$

Step 4.5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.