Algebra Examples

$f (x) = - 12 \sqrt[3]{x}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 12 \sqrt[3]{x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 12 \sqrt[3]{x} = 0$ .

$- 12 \sqrt[3]{x} = 0$

Step 1.2.2

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

${(- 12 \sqrt[3]{x})}^{3} = 0^{3}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

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

${(- 12 x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify ${(- 12 x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.3.2.1.1

Apply the product rule to $- 12 x^{\frac{1}{3}}$ .

${(- 12)}^{3} {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.3.2.1.2

Raise $- 12$ to the power of $3$ .

$- 1728 {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.3.2.1.3

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

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Step 1.2.3.2.1.3.1

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

$- 1728 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.3.2.1.3.2

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.3.2.1

Cancel the common factor.

$- 1728 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.3.2.1.3.2.2

Rewrite the expression.

$- 1728 x^{1} = 0^{3}$

Step 1.2.3.2.1.4

Simplify.

$- 1728 x = 0^{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Raising $0$ to any positive power yields $0$ .

$- 1728 x = 0$

Step 1.2.4

Divide each term in $- 1728 x = 0$ by $- 1728$ and simplify.

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Step 1.2.4.1

Divide each term in $- 1728 x = 0$ by $- 1728$ .

$\frac{- 1728 x}{- 1728} = \frac{0}{- 1728}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $- 1728$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{- 1728 x}{- 1728} = \frac{0}{- 1728}$

Step 1.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1728}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $0$ by $- 1728$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 12 \sqrt[3]{0}$

Step 2.2

Simplify $- 12 \sqrt[3]{0}$ .

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Step 2.2.1

Remove parentheses.

$y = - 12 \sqrt[3]{0}$

Step 2.2.2

Rewrite $0$ as $0^{3}$ .

$y = - 12 \sqrt[3]{0^{3}}$

Step 2.2.3

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

$y = - 12 \cdot 0$

Step 2.2.4

Multiply $- 12$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0)$

y-intercept(s): $(0, 0)$

Step 4