Precalculus Examples

$\frac{6 x^{2}}{5 y} \div \frac{3 x}{10 y^{2}}$

Step 1

Set the denominator in $\frac{6 x^{2}}{5 y}$ equal to $0$ to find where the expression is undefined.

$5 y = 0$

Step 2

Divide each term in $5 y = 0$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 y = 0$ by $5$ .

$\frac{5 y}{5} = \frac{0}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{0}{5}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $5$ .

$y = 0$

Step 3

Set the denominator in $\frac{3 x}{10 y^{2}}$ equal to $0$ to find where the expression is undefined.

$10 y^{2} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

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

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

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.

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

Step 4.1.2.1.2

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

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

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $10$ .

$y^{2} = 0$

Step 4.2

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

$y = \pm \sqrt{0}$

Step 4.3

Simplify $\pm \sqrt{0}$ .

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Step 4.3.1

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

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

Step 4.3.2

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

$y = \pm 0$

Step 4.3.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 5

Set the denominator in $\frac{6 x^{2}}{5 y} \div \frac{3 x}{10 y^{2}}$ equal to $0$ to find where the expression is undefined.

$\frac{3 x}{10 y^{2}} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$3 x = 0$

Step 6.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $0$ by $3$ .

$x = 0$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$