Pre-Algebra Examples

$2 p r^{2} + \frac{67.2}{r}$

Step 1

Simplify.

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Step 1.1

Subtract $\frac{67.2}{x}$ from both sides of the equation.

$2 y x^{2} = - \frac{67.2}{x}$

Step 1.2

Divide each term in $2 y x^{2} = - \frac{67.2}{x}$ by $2 x^{2}$ and simplify.

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Step 1.2.1

Divide each term in $2 y x^{2} = - \frac{67.2}{x}$ by $2 x^{2}$ .

$\frac{2 y x^{2}}{2 x^{2}} = \frac{- \frac{67.2}{x}}{2 x^{2}}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 y x^{2}}{2 x^{2}} = \frac{- \frac{67.2}{x}}{2 x^{2}}$

Step 1.2.2.1.2

Rewrite the expression.

$\frac{y x^{2}}{x^{2}} = \frac{- \frac{67.2}{x}}{2 x^{2}}$

Step 1.2.2.2

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

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Step 1.2.2.2.1

Cancel the common factor.

$\frac{y x^{2}}{x^{2}} = \frac{- \frac{67.2}{x}}{2 x^{2}}$

Step 1.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- \frac{67.2}{x}}{2 x^{2}}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{67.2}{x} \cdot \frac{1}{2 x^{2}}$

Step 1.2.3.2

Multiply $- \frac{67.2}{x} \cdot \frac{1}{2 x^{2}}$ .

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Step 1.2.3.2.1

Multiply $\frac{1}{2 x^{2}}$ by $\frac{67.2}{x}$ .

$y = - \frac{67.2}{2 x^{2} x}$

Step 1.2.3.2.2

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

$y = - \frac{67.2}{2 (x^{1} x^{2})}$

Step 1.2.3.2.3

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

$y = - \frac{67.2}{2 x^{1 + 2}}$

Step 1.2.3.2.4

Add $1$ and $2$ .

$y = - \frac{67.2}{2 x^{3}}$

Step 1.2.3.3

Factor $67.2$ out of $67.2$ .

$y = - \frac{67.2 (1)}{2 x^{3}}$

Step 1.2.3.4

Factor $2$ out of $2 x^{3}$ .

$y = - \frac{67.2 (1)}{2 (x^{3})}$

Step 1.2.3.5

Separate fractions.

$y = - (\frac{67.2}{2} \cdot \frac{1}{x^{3}})$

Step 1.2.3.6

Divide $67.2$ by $2$ .

$y = - (33.6 \frac{1}{x^{3}})$

Step 1.2.3.7

Combine $33.6$ and $\frac{1}{x^{3}}$ .

$y = - \frac{33.6}{x^{3}}$

Step 2

Find where the expression $- \frac{33.6}{x^{3}}$ is undefined.

$x = 0$

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find $n$ and $m$ .

$n = 0$

$m = 3$

Step 5

Since $n < m$ , the x-axis, $y = 0$ , is the horizontal asymptote.

$y = 0$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

No Oblique Asymptotes

Step 8