Precalculus Examples

$\frac{6 a x}{m^{2} - 2 m} \div \frac{8 a x}{3 m - 6}$

Step 1

Set the denominator in $\frac{6 a x}{m^{2} - 2 m}$ equal to $0$ to find where the expression is undefined.

$m^{2} - 2 m = 0$

Step 2

Solve for $a$ .

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Step 2.1

Factor $m$ out of $m^{2} - 2 m$ .

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Step 2.1.1

Factor $m$ out of $m^{2}$ .

$m \cdot m - 2 m = 0$

Step 2.1.2

Factor $m$ out of $- 2 m$ .

$m \cdot m + m \cdot - 2 = 0$

Step 2.1.3

Factor $m$ out of $m \cdot m + m \cdot - 2$ .

$m (m - 2) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$m = 0$

$m - 2 = 0$

Step 2.3

Set $m$ equal to $0$ .

$m = 0$

Step 2.4

Set $m - 2$ equal to $0$ and solve for $m$ .

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Step 2.4.1

Set $m - 2$ equal to $0$ .

$m - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$m = 2$

Step 2.5

The final solution is all the values that make $m (m - 2) = 0$ true.

$m = 0, 2$

Step 3

Set the denominator in $\frac{8 a x}{3 m - 6}$ equal to $0$ to find where the expression is undefined.

$3 m - 6 = 0$

Step 4

Solve for $a$ .

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Step 4.1

Add $6$ to both sides of the equation.

$3 m = 6$

Step 4.2

Divide each term in $3 m = 6$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 m = 6$ by $3$ .

$\frac{3 m}{3} = \frac{6}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 m}{3} = \frac{6}{3}$

Step 4.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{6}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $6$ by $3$ .

$m = 2$

Step 5

Set the denominator in $\frac{6 a x}{m^{2} - 2 m} \div \frac{8 a x}{3 m - 6}$ equal to $0$ to find where the expression is undefined.

$\frac{8 a x}{3 m - 6} = 0$

Step 6

Solve for $a$ .

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Step 6.1

Set the numerator equal to zero.

$8 a x = 0$

Step 6.2

Divide each term in $8 a x = 0$ by $8 x$ and simplify.

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Step 6.2.1

Divide each term in $8 a x = 0$ by $8 x$ .

$\frac{8 a x}{8 x} = \frac{0}{8 x}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $8$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{8 a x}{8 x} = \frac{0}{8 x}$

Step 6.2.2.1.2

Rewrite the expression.

$\frac{a x}{x} = \frac{0}{8 x}$

Step 6.2.2.2

Cancel the common factor of $x$ .

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{a x}{x} = \frac{0}{8 x}$

Step 6.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{0}{8 x}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $0$ and $8$ .

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Step 6.2.3.1.1

Factor $8$ out of $0$ .

$a = \frac{8 (0)}{8 x}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $8$ out of $8 x$ .

$a = \frac{8 (0)}{8 (x)}$

Step 6.2.3.1.2.2

Cancel the common factor.

$a = \frac{8 \cdot 0}{8 x}$

Step 6.2.3.1.2.3

Rewrite the expression.

$a = \frac{0}{x}$

Step 6.2.3.2

Divide $0$ by $x$ .

$a = 0$

Step 7

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

Interval Notation:

$(- \infty, 0) \cup (0, 2) \cup (2, \infty)$

Set-Builder Notation:

${a | a \neq 0, 2}$