Precalculus Examples

$(m^{2} - 16 n^{2}) \div \frac{3 m + 12 n}{m n}$

Step 1

Set the denominator in $\frac{3 m + 12 n}{m n}$ equal to $0$ to find where the expression is undefined.

$m n = 0$

Step 2

Divide each term in $m n = 0$ by $n$ and simplify.

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Step 2.1

Divide each term in $m n = 0$ by $n$ .

$\frac{m n}{n} = \frac{0}{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $n$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{m n}{n} = \frac{0}{n}$

Step 2.2.1.2

Divide $m$ by $1$ .

$m = \frac{0}{n}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $n$ .

$m = 0$

Step 3

Set the denominator in $(m^{2} - 16 n^{2}) \div \frac{3 m + 12 n}{m n}$ equal to $0$ to find where the expression is undefined.

$\frac{3 m + 12 n}{m n} = 0$

Step 4

Solve for $m$ .

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Step 4.1

Set the numerator equal to zero.

$3 m + 12 n = 0$

Step 4.2

Solve the equation for $m$ .

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Step 4.2.1

Subtract $12 n$ from both sides of the equation.

$3 m = - 12 n$

Step 4.2.2

Divide each term in $3 m = - 12 n$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 m = - 12 n$ by $3$ .

$\frac{3 m}{3} = \frac{- 12 n}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 m}{3} = \frac{- 12 n}{3}$

Step 4.2.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 12 n}{3}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Cancel the common factor of $- 12$ and $3$ .

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Step 4.2.2.3.1.1

Factor $3$ out of $- 12 n$ .

$m = \frac{3 (- 4 n)}{3}$

Step 4.2.2.3.1.2

Cancel the common factors.

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Step 4.2.2.3.1.2.1

Factor $3$ out of $3$ .

$m = \frac{3 (- 4 n)}{3 (1)}$

Step 4.2.2.3.1.2.2

Cancel the common factor.

$m = \frac{3 (- 4 n)}{3 \cdot 1}$

Step 4.2.2.3.1.2.3

Rewrite the expression.

$m = \frac{- 4 n}{1}$

Step 4.2.2.3.1.2.4

Divide $- 4 n$ by $1$ .

$m = - 4 n$

Step 5

The domain is all values of $m$ that make the expression defined.

Set-Builder Notation:

${m | m \neq 0, m \neq - 4 n}$ , for any integer $n$