Finite Math Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 x^{3} - 2 x^{- 2} = 0$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 x^{3} - 2 \frac{1}{x^{2}} = 0$

Step 1.2.2.2

Combine $- 2$ and $\frac{1}{x^{2}}$ .

$3 x^{3} + \frac{- 2}{x^{2}} = 0$

Step 1.2.2.3

Move the negative in front of the fraction.

$3 x^{3} - \frac{2}{x^{2}} = 0$

Step 1.2.3

Find the LCD of the terms in the equation.

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Step 1.2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1$

Step 1.2.3.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 1.2.4

Multiply each term in $3 x^{3} - \frac{2}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 1.2.4.1

Multiply each term in $3 x^{3} - \frac{2}{x^{2}} = 0$ by $x^{2}$ .

$3 x^{3} x^{2} - \frac{2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 1.2.4.2.1.1.1

Move $x^{2}$ .

$3 (x^{2} x^{3}) - \frac{2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2.1.1.2

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

$3 x^{2 + 3} - \frac{2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2.1.1.3

Add $2$ and $3$ .

$3 x^{5} - \frac{2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 1.2.4.2.1.2.1

Move the leading negative in $- \frac{2}{x^{2}}$ into the numerator.

$3 x^{5} + \frac{- 2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2.1.2.2

Cancel the common factor.

$3 x^{5} + \frac{- 2}{x^{2}} x^{2} = 0 x^{2}$

Step 1.2.4.2.1.2.3

Rewrite the expression.

$3 x^{5} - 2 = 0 x^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Multiply $0$ by $x^{2}$ .

$3 x^{5} - 2 = 0$

Step 1.2.5

Solve the equation.

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Step 1.2.5.1

Add $2$ to both sides of the equation.

$3 x^{5} = 2$

Step 1.2.5.2

Divide each term in $3 x^{5} = 2$ by $3$ and simplify.

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Step 1.2.5.2.1

Divide each term in $3 x^{5} = 2$ by $3$ .

$\frac{3 x^{5}}{3} = \frac{2}{3}$

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 $3$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{3 x^{5}}{3} = \frac{2}{3}$

Step 1.2.5.2.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{2}{3}$

Step 1.2.5.3

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

$x = \sqrt[5]{\frac{2}{3}}$

Step 1.2.5.4

Simplify $\sqrt[5]{\frac{2}{3}}$ .

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Step 1.2.5.4.1

Rewrite $\sqrt[5]{\frac{2}{3}}$ as $\frac{\sqrt[5]{2}}{\sqrt[5]{3}}$ .

$x = \frac{\sqrt[5]{2}}{\sqrt[5]{3}}$

Step 1.2.5.4.2

Multiply $\frac{\sqrt[5]{2}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$x = \frac{\sqrt[5]{2}}{\sqrt[5]{3}} \cdot \frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$

Step 1.2.5.4.3

Combine and simplify the denominator.

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Step 1.2.5.4.3.1

Multiply $\frac{\sqrt[5]{2}}{\sqrt[5]{3}}$ by $\frac{{\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{4}}$ .

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{\sqrt[5]{3} {\sqrt[5]{3}}^{4}}$

Step 1.2.5.4.3.2

Raise $\sqrt[5]{3}$ to the power of $1$ .

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1} {\sqrt[5]{3}}^{4}}$

Step 1.2.5.4.3.3

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

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{1 + 4}}$

Step 1.2.5.4.3.4

Add $1$ and $4$ .

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{{\sqrt[5]{3}}^{5}}$

Step 1.2.5.4.3.5

Rewrite ${\sqrt[5]{3}}^{5}$ as $3$ .

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Step 1.2.5.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3}$ as $3^{\frac{1}{5}}$ .

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{{(3^{\frac{1}{5}})}^{5}}$

Step 1.2.5.4.3.5.2

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

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{3^{\frac{1}{5} \cdot 5}}$

Step 1.2.5.4.3.5.3

Combine $\frac{1}{5}$ and $5$ .

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 1.2.5.4.3.5.4

Cancel the common factor of $5$ .

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Step 1.2.5.4.3.5.4.1

Cancel the common factor.

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{3^{\frac{5}{5}}}$

Step 1.2.5.4.3.5.4.2

Rewrite the expression.

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{3^{1}}$

Step 1.2.5.4.3.5.5

Evaluate the exponent.

$x = \frac{\sqrt[5]{2} {\sqrt[5]{3}}^{4}}{3}$

Step 1.2.5.4.4

Simplify the numerator.

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Step 1.2.5.4.4.1

Rewrite ${\sqrt[5]{3}}^{4}$ as $\sqrt[5]{3^{4}}$ .

$x = \frac{\sqrt[5]{2} \sqrt[5]{3^{4}}}{3}$

Step 1.2.5.4.4.2

Raise $3$ to the power of $4$ .

$x = \frac{\sqrt[5]{2} \sqrt[5]{81}}{3}$

Step 1.2.5.4.5

Simplify the numerator.

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Step 1.2.5.4.5.1

Combine using the product rule for radicals.

$x = \frac{\sqrt[5]{2 \cdot 81}}{3}$

Step 1.2.5.4.5.2

Multiply $2$ by $81$ .

$x = \frac{\sqrt[5]{162}}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt[5]{162}}{3}, 0)$

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 $y$ .

$y = 3 {(0)}^{3} - 2 {(0)}^{- 2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \cdot 0^{3} - 2 {(0)}^{- 2}$

Step 2.2.2

Remove parentheses.

$y = 3 \cdot 0^{3} - 2 \cdot 0^{- 2}$

Step 2.2.3

Remove parentheses.

$y = 3 {(0)}^{3} - 2 {(0)}^{- 2}$

Step 2.2.4

Simplify the right side.

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 3 \cdot 0 - 2 {(0)}^{- 2}$

Step 2.2.4.1.2

Multiply $3$ by $0$ .

$y = 0 - 2 {(0)}^{- 2}$

Step 2.2.4.1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = 0 - 2 \frac{1}{0^{2}}$

Step 2.2.4.1.4

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

$y = 0 - 2 (\frac{1}{0})$

Step 2.2.4.2

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(\frac{\sqrt[5]{162}}{3}, 0)$

y-intercept(s): $None$

Step 4