Precalculus Examples

$\frac{y - 1}{y \sqrt{y + 1}}$

Step 1

Set the radicand in $\sqrt{y + 1}$ greater than or equal to $0$ to find where the expression is defined.

$y + 1 \geq 0$

Step 2

Subtract $1$ from both sides of the inequality.

$y \geq - 1$

Step 3

Set the denominator in $\frac{y - 1}{y \sqrt{y + 1}}$ equal to $0$ to find where the expression is undefined.

$y \sqrt{y + 1} = 0$

Step 4

Solve for $y$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(y \sqrt{y + 1})}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 1}$ as ${(y + 1)}^{\frac{1}{2}}$ .

${(y {(y + 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${(y {(y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Apply the product rule to $y {(y + 1)}^{\frac{1}{2}}$ .

$y^{2} {({(y + 1)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2.1.2

Multiply the exponents in ${({(y + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{2} {(y + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.2.1

Cancel the common factor.

$y^{2} {(y + 1)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.2.2.2

Rewrite the expression.

$y^{2} {(y + 1)}^{1} = 0^{2}$

Step 4.2.2.1.3

Simplify.

$y^{2} (y + 1) = 0^{2}$

Step 4.2.2.1.4

Apply the distributive property.

$y^{2} y + y^{2} \cdot 1 = 0^{2}$

Step 4.2.2.1.5

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 4.2.2.1.5.1

Multiply $y^{2}$ by $y$ .

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Step 4.2.2.1.5.1.1

Raise $y$ to the power of $1$ .

$y^{2} y^{1} + y^{2} \cdot 1 = 0^{2}$

Step 4.2.2.1.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2 + 1} + y^{2} \cdot 1 = 0^{2}$

Step 4.2.2.1.5.2

Add $2$ and $1$ .

$y^{3} + y^{2} \cdot 1 = 0^{2}$

Step 4.2.2.1.6

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

$y^{3} + y^{2} = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$y^{3} + y^{2} = 0$

Step 4.3

Solve for $y$ .

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Step 4.3.1

Factor $y^{2}$ out of $y^{3} + y^{2}$ .

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Step 4.3.1.1

Factor $y^{2}$ out of $y^{3}$ .

$y^{2} y + y^{2} = 0$

Step 4.3.1.2

Multiply by $1$ .

$y^{2} y + y^{2} \cdot 1 = 0$

Step 4.3.1.3

Factor $y^{2}$ out of $y^{2} y + y^{2} \cdot 1$ .

$y^{2} (y + 1) = 0$

Step 4.3.2

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

$y^{2} = 0$

$y + 1 = 0$

Step 4.3.3

Set $y^{2}$ equal to $0$ and solve for $y$ .

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Step 4.3.3.1

Set $y^{2}$ equal to $0$ .

$y^{2} = 0$

Step 4.3.3.2

Solve $y^{2} = 0$ for $y$ .

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Step 4.3.3.2.1

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

$y = \pm \sqrt{0}$

Step 4.3.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.3.3.2.2.1

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

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

Step 4.3.3.2.2.2

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

$y = \pm 0$

Step 4.3.3.2.2.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 4.3.4

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 4.3.4.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 4.3.4.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 4.3.5

The final solution is all the values that make $y^{2} (y + 1) = 0$ true.

$y = 0, - 1$

Step 5

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

Interval Notation:

$(- 1, 0) \cup (0, \infty)$

Set-Builder Notation:

${y | y > - 1, y \neq 0}$

Step 6