Precalculus Examples

$\frac{2 a b}{a^{2} - b^{2}} \div \frac{4 a b}{{(a - b)}^{2}}$

Step 1

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

$a^{2} - b^{2} = 0$

Step 2

Solve for $a$ .

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Step 2.1

Add $b^{2}$ to both sides of the equation.

$a^{2} = b^{2}$

Step 2.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| a | = | b |$

Step 2.3

Solve for $a$ .

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Step 2.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$a = \pm | b |$

Step 2.3.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.2.1

First, use the positive value of the $\pm$ to find the first solution.

$a = | b |$

Step 2.3.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - | b |$

Step 2.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

$a = | b |$

$a = - | b |$

Step 3

Set the denominator in $\frac{4 a b}{{(a - b)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(a - b)}^{2} = 0$

Step 4

Solve for $a$ .

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Step 4.1

Set the $a - b$ equal to $0$ .

$a - b = 0$

Step 4.2

Add $b$ to both sides of the equation.

$a = b$

Step 5

Set the denominator in $\frac{2 a b}{a^{2} - b^{2}} \div \frac{4 a b}{{(a - b)}^{2}}$ equal to $0$ to find where the expression is undefined.

$\frac{4 a b}{{(a - b)}^{2}} = 0$

Step 6

Solve for $a$ .

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Step 6.1

Set the numerator equal to zero.

$4 a b = 0$

Step 6.2

Divide each term in $4 a b = 0$ by $4 b$ and simplify.

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Step 6.2.1

Divide each term in $4 a b = 0$ by $4 b$ .

$\frac{4 a b}{4 b} = \frac{0}{4 b}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{4 a b}{4 b} = \frac{0}{4 b}$

Step 6.2.2.1.2

Rewrite the expression.

$\frac{a b}{b} = \frac{0}{4 b}$

Step 6.2.2.2

Cancel the common factor of $b$ .

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{a b}{b} = \frac{0}{4 b}$

Step 6.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{0}{4 b}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Cancel the common factor of $0$ and $4$ .

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Step 6.2.3.1.1

Factor $4$ out of $0$ .

$a = \frac{4 (0)}{4 b}$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $4$ out of $4 b$ .

$a = \frac{4 (0)}{4 (b)}$

Step 6.2.3.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot 0}{4 b}$

Step 6.2.3.1.2.3

Rewrite the expression.

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

Step 6.2.3.2

Divide $0$ by $b$ .

$a = 0$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$