Algebra Examples

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

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = 3 \sqrt[3]{- 2 x} + 1$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $3 \sqrt[3]{- 2 x} + 1 = 0$ .

$3 \sqrt[3]{- 2 x} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$3 \sqrt[3]{- 2 x} = - 1$

Step 1.2.3

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

${(3 \sqrt[3]{- 2 x})}^{3} = {(- 1)}^{3}$

Step 1.2.4

Simplify each side of the equation.

Tap for more steps...

Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

Tap for more steps...

Step 1.2.4.2.1

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

Tap for more steps...

Step 1.2.4.2.1.1

Apply the product rule to $- 2 x$ .

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

Step 1.2.4.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 1.2.4.2.1.2.1

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

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

Step 1.2.4.2.1.2.2

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

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

Step 1.2.4.2.1.3

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

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

Step 1.2.4.2.1.4

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

Tap for more steps...

Step 1.2.4.2.1.4.1

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

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

Step 1.2.4.2.1.4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.2.1.4.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.4.2.2

Rewrite the expression.

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

Step 1.2.4.2.1.5

Evaluate the exponent.

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

Step 1.2.4.2.1.6

Multiply $27$ by $- 2$ .

$- 54 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 1.2.4.2.1.7

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

Tap for more steps...

Step 1.2.4.2.1.7.1

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

$- 54 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 1.2.4.2.1.7.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.2.1.7.2.1

Cancel the common factor.

$- 54 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 1.2.4.2.1.7.2.2

Rewrite the expression.

$- 54 x^{1} = {(- 1)}^{3}$

Step 1.2.4.2.1.8

Simplify.

$- 54 x = {(- 1)}^{3}$

Step 1.2.4.3

Simplify the right side.

Tap for more steps...

Step 1.2.4.3.1

Raise $- 1$ to the power of $3$ .

$- 54 x = - 1$

Step 1.2.5

Divide each term in $- 54 x = - 1$ by $- 54$ and simplify.

Tap for more steps...

Step 1.2.5.1

Divide each term in $- 54 x = - 1$ by $- 54$ .

$\frac{- 54 x}{- 54} = \frac{- 1}{- 54}$

Step 1.2.5.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.2.1

Cancel the common factor of $- 54$ .

Tap for more steps...

Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{- 54 x}{- 54} = \frac{- 1}{- 54}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 54}$

Step 1.2.5.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{54}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{54}, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = 3 \sqrt[3]{- 2 \cdot 0} + 1$

Step 2.2

Simplify $3 \sqrt[3]{- 2 \cdot 0} + 1$ .

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $- 2$ by $0$ .

$y = 3 \sqrt[3]{0} + 1$

Step 2.2.1.2

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

$y = 3 \sqrt[3]{0^{3}} + 1$

Step 2.2.1.3

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

$y = 3 \cdot 0 + 1$

Step 2.2.1.4

Multiply $3$ by $0$ .

$y = 0 + 1$

Step 2.2.2

Add $0$ and $1$ .

$y = 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{1}{54}, 0)$

y-intercept(s): $(0, 1)$

Step 4