Precalculus Examples

$\frac{3 a b}{4 x y} \div (- \frac{21 a^{2} b}{10 x^{2} y})$

Step 1

Set the denominator in $\frac{3 a b}{4 x y}$ equal to $0$ to find where the expression is undefined.

$4 x y = 0$

Step 2

Divide each term in $4 x y = 0$ by $4 y$ and simplify.

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Step 2.1

Divide each term in $4 x y = 0$ by $4 y$ .

$\frac{4 x y}{4 y} = \frac{0}{4 y}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 x y}{4 y} = \frac{0}{4 y}$

Step 2.2.1.2

Rewrite the expression.

$\frac{x y}{y} = \frac{0}{4 y}$

Step 2.2.2

Cancel the common factor of $y$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{x y}{y} = \frac{0}{4 y}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{4 y}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Factor $4$ out of $0$ .

$x = \frac{4 (0)}{4 y}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $4$ out of $4 y$ .

$x = \frac{4 (0)}{4 (y)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 0}{4 y}$

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Divide $0$ by $y$ .

$x = 0$

Step 3

Set the denominator in $\frac{21 a^{2} b}{10 x^{2} y}$ equal to $0$ to find where the expression is undefined.

$10 x^{2} y = 0$

Step 4

Solve for $a$ .

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Step 4.1

Divide each term in $10 x^{2} y = 0$ by $10 y$ and simplify.

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Step 4.1.1

Divide each term in $10 x^{2} y = 0$ by $10 y$ .

$\frac{10 x^{2} y}{10 y} = \frac{0}{10 y}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $10$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{10 x^{2} y}{10 y} = \frac{0}{10 y}$

Step 4.1.2.1.2

Rewrite the expression.

$\frac{x^{2} y}{y} = \frac{0}{10 y}$

Step 4.1.2.2

Cancel the common factor of $y$ .

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Step 4.1.2.2.1

Cancel the common factor.

$\frac{x^{2} y}{y} = \frac{0}{10 y}$

Step 4.1.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{10 y}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Cancel the common factor of $0$ and $10$ .

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Step 4.1.3.1.1

Factor $10$ out of $0$ .

$x^{2} = \frac{10 (0)}{10 y}$

Step 4.1.3.1.2

Cancel the common factors.

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Step 4.1.3.1.2.1

Factor $10$ out of $10 y$ .

$x^{2} = \frac{10 (0)}{10 (y)}$

Step 4.1.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{10 \cdot 0}{10 y}$

Step 4.1.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{0}{y}$

Step 4.1.3.2

Divide $0$ by $y$ .

$x^{2} = 0$

Step 4.2

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

$x = \pm \sqrt{0}$

Step 4.3

Simplify $\pm \sqrt{0}$ .

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Step 4.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Set the denominator in $\frac{3 a b}{4 x y} \div (- \frac{21 a^{2} b}{10 x^{2} y})$ equal to $0$ to find where the expression is undefined.

$- \frac{21 a^{2} b}{10 x^{2} y} = 0$

Step 6

Solve for $a$ .

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Step 6.1

Set the numerator equal to zero.

$21 a^{2} b = 0$

Step 6.2

Solve the equation for $a$ .

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Step 6.2.1

Divide each term in $21 a^{2} b = 0$ by $21 b$ and simplify.

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Step 6.2.1.1

Divide each term in $21 a^{2} b = 0$ by $21 b$ .

$\frac{21 a^{2} b}{21 b} = \frac{0}{21 b}$

Step 6.2.1.2

Simplify the left side.

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Step 6.2.1.2.1

Cancel the common factor of $21$ .

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Step 6.2.1.2.1.1

Cancel the common factor.

$\frac{21 a^{2} b}{21 b} = \frac{0}{21 b}$

Step 6.2.1.2.1.2

Rewrite the expression.

$\frac{a^{2} b}{b} = \frac{0}{21 b}$

Step 6.2.1.2.2

Cancel the common factor of $b$ .

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Step 6.2.1.2.2.1

Cancel the common factor.

$\frac{a^{2} b}{b} = \frac{0}{21 b}$

Step 6.2.1.2.2.2

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{0}{21 b}$

Step 6.2.1.3

Simplify the right side.

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Step 6.2.1.3.1

Cancel the common factor of $0$ and $21$ .

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Step 6.2.1.3.1.1

Factor $21$ out of $0$ .

$a^{2} = \frac{21 (0)}{21 b}$

Step 6.2.1.3.1.2

Cancel the common factors.

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Step 6.2.1.3.1.2.1

Factor $21$ out of $21 b$ .

$a^{2} = \frac{21 (0)}{21 (b)}$

Step 6.2.1.3.1.2.2

Cancel the common factor.

$a^{2} = \frac{21 \cdot 0}{21 b}$

Step 6.2.1.3.1.2.3

Rewrite the expression.

$a^{2} = \frac{0}{b}$

Step 6.2.1.3.2

Divide $0$ by $b$ .

$a^{2} = 0$

Step 6.2.2

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

$a = \pm \sqrt{0}$

Step 6.2.3

Simplify $\pm \sqrt{0}$ .

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Step 6.2.3.1

Rewrite $0$ as $0^{2}$ .

$a = \pm \sqrt{0^{2}}$

Step 6.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$a = \pm 0$

Step 6.2.3.3

Plus or minus $0$ is $0$ .

$a = 0$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$