Algebra Examples

$f (x) = 3 \sqrt[3]{- x - 2} + 6$

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 $x$ .

$0 = 3 \sqrt[3]{- x - 2} + 6$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 \sqrt[3]{- x - 2} + 6 = 0$ .

$3 \sqrt[3]{- x - 2} + 6 = 0$

Step 1.2.2

Subtract $6$ from both sides of the equation.

$3 \sqrt[3]{- x - 2} = - 6$

Step 1.2.3

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

${(3 \sqrt[3]{- x - 2})}^{3} = {(- 6)}^{3}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{- x - 2}$ as ${(- x - 2)}^{\frac{1}{3}}$ .

${(3 {(- x - 2)}^{\frac{1}{3}})}^{3} = {(- 6)}^{3}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(3 {(- x - 2)}^{\frac{1}{3}})}^{3}$ .

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Step 1.2.4.2.1.1

Apply the product rule to $3 {(- x - 2)}^{\frac{1}{3}}$ .

$3^{3} {({(- x - 2)}^{\frac{1}{3}})}^{3} = {(- 6)}^{3}$

Step 1.2.4.2.1.2

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

$27 {({(- x - 2)}^{\frac{1}{3}})}^{3} = {(- 6)}^{3}$

Step 1.2.4.2.1.3

Multiply the exponents in ${({(- x - 2)}^{\frac{1}{3}})}^{3}$ .

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Step 1.2.4.2.1.3.1

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

$27 {(- x - 2)}^{\frac{1}{3} \cdot 3} = {(- 6)}^{3}$

Step 1.2.4.2.1.3.2

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.3.2.1

Cancel the common factor.

$27 {(- x - 2)}^{\frac{1}{3} \cdot 3} = {(- 6)}^{3}$

Step 1.2.4.2.1.3.2.2

Rewrite the expression.

$27 {(- x - 2)}^{1} = {(- 6)}^{3}$

Step 1.2.4.2.1.4

Simplify.

$27 (- x - 2) = {(- 6)}^{3}$

Step 1.2.4.2.1.5

Apply the distributive property.

$27 (- x) + 27 \cdot - 2 = {(- 6)}^{3}$

Step 1.2.4.2.1.6

Multiply.

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $27$ .

$- 27 x + 27 \cdot - 2 = {(- 6)}^{3}$

Step 1.2.4.2.1.6.2

Multiply $27$ by $- 2$ .

$- 27 x - 54 = {(- 6)}^{3}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Raise $- 6$ to the power of $3$ .

$- 27 x - 54 = - 216$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.1.1

Add $54$ to both sides of the equation.

$- 27 x = - 216 + 54$

Step 1.2.5.1.2

Add $- 216$ and $54$ .

$- 27 x = - 162$

Step 1.2.5.2

Divide each term in $- 27 x = - 162$ by $- 27$ and simplify.

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Step 1.2.5.2.1

Divide each term in $- 27 x = - 162$ by $- 27$ .

$\frac{- 27 x}{- 27} = \frac{- 162}{- 27}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $- 27$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{- 27 x}{- 27} = \frac{- 162}{- 27}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 162}{- 27}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $- 162$ by $- 27$ .

$x = 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0)$

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 $y$ .

$y = 3 \sqrt[3]{- (0) - 2} + 6$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $0$ .

$y = 3 \sqrt[3]{0 - 2} + 6$

Step 2.2.2

Subtract $2$ from $0$ .

$y = 3 \sqrt[3]{- 2} + 6$

Step 2.2.3

Rewrite $- 2$ as ${(- 1)}^{3} \cdot 2$ .

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Step 2.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$y = 3 \sqrt[3]{- 1 (2)} + 6$

Step 2.2.3.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = 3 \sqrt[3]{{(- 1)}^{3} \cdot 2} + 6$

Step 2.2.4

Pull terms out from under the radical.

$y = 3 (- 1 \sqrt[3]{2}) + 6$

Step 2.2.5

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$y = 3 (- \sqrt[3]{2}) + 6$

Step 2.2.6

Multiply $- 1$ by $3$ .

$y = - 3 \sqrt[3]{2} + 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3 \sqrt[3]{2} + 6)$

Step 3

List the intersections.

x-intercept(s): $(6, 0)$

y-intercept(s): $(0, - 3 \sqrt[3]{2} + 6)$

Step 4