Precalculus Examples

$\frac{1}{\sqrt{y}} (\frac{1}{\sqrt{x} - \sqrt{y}} - \frac{1}{\sqrt{x} + \sqrt{y}})$

Step 1

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

$y \geq 0$

Step 2

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

$x \geq 0$

Step 3

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

$\sqrt{y} = 0$

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.

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

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 $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

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

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.2.1.1.2.2

Rewrite the expression.

$y^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$y = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$y = 0$

Step 5

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

$\sqrt{x} - \sqrt{y} = 0$

Step 6

Solve for $y$ .

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Step 6.1

Subtract $\sqrt{x}$ from both sides of the equation.

$- \sqrt{y} = - \sqrt{x}$

Step 6.2

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

${(- \sqrt{y})}^{2} = {(- \sqrt{x})}^{2}$

Step 6.3

Simplify each side of the equation.

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Step 6.3.1

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

${(- y^{\frac{1}{2}})}^{2} = {(- \sqrt{x})}^{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

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

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Step 6.3.2.1.1

Apply the product rule to $- y^{\frac{1}{2}}$ .

${(- 1)}^{2} {(y^{\frac{1}{2}})}^{2} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {(y^{\frac{1}{2}})}^{2} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.3

Multiply ${(y^{\frac{1}{2}})}^{2}$ by $1$ .

${(y^{\frac{1}{2}})}^{2} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.4

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

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Step 6.3.2.1.4.1

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

$y^{\frac{1}{2} \cdot 2} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.4.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.4.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.4.2.2

Rewrite the expression.

$y^{1} = {(- \sqrt{x})}^{2}$

Step 6.3.2.1.5

Simplify.

$y = {(- \sqrt{x})}^{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify ${(- \sqrt{x})}^{2}$ .

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Step 6.3.3.1.1

Simplify the expression.

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Step 6.3.3.1.1.1

Apply the product rule to $- \sqrt{x}$ .

$y = {(- 1)}^{2} {\sqrt{x}}^{2}$

Step 6.3.3.1.1.2

Raise $- 1$ to the power of $2$ .

$y = 1 {\sqrt{x}}^{2}$

Step 6.3.3.1.1.3

Multiply ${\sqrt{x}}^{2}$ by $1$ .

$y = {\sqrt{x}}^{2}$

Step 6.3.3.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 6.3.3.1.2.1

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

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

Step 6.3.3.1.2.2

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

$y = x^{\frac{1}{2} \cdot 2}$

Step 6.3.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 6.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 6.3.3.1.2.4.1

Cancel the common factor.

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

Step 6.3.3.1.2.4.2

Rewrite the expression.

$y = x^{1}$

Step 6.3.3.1.2.5

Simplify.

$y = x$

Step 7

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

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

Step 8

Solve for $y$ .

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Step 8.1

Subtract $\sqrt{x}$ from both sides of the equation.

$\sqrt{y} = - \sqrt{x}$

Step 8.2

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

${\sqrt{y}}^{2} = {(- \sqrt{x})}^{2}$

Step 8.3

Simplify each side of the equation.

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Step 8.3.1

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

${(y^{\frac{1}{2}})}^{2} = {(- \sqrt{x})}^{2}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

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

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Step 8.3.2.1.1

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

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Step 8.3.2.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(- \sqrt{x})}^{2}$

Step 8.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 8.3.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(- \sqrt{x})}^{2}$

Step 8.3.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(- \sqrt{x})}^{2}$

Step 8.3.2.1.2

Simplify.

$y = {(- \sqrt{x})}^{2}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Simplify ${(- \sqrt{x})}^{2}$ .

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Step 8.3.3.1.1

Simplify the expression.

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Step 8.3.3.1.1.1

Apply the product rule to $- \sqrt{x}$ .

$y = {(- 1)}^{2} {\sqrt{x}}^{2}$

Step 8.3.3.1.1.2

Raise $- 1$ to the power of $2$ .

$y = 1 {\sqrt{x}}^{2}$

Step 8.3.3.1.1.3

Multiply ${\sqrt{x}}^{2}$ by $1$ .

$y = {\sqrt{x}}^{2}$

Step 8.3.3.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 8.3.3.1.2.1

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

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

Step 8.3.3.1.2.2

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

$y = x^{\frac{1}{2} \cdot 2}$

Step 8.3.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 8.3.3.1.2.4

Cancel the common factor of $2$ .

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Step 8.3.3.1.2.4.1

Cancel the common factor.

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

Step 8.3.3.1.2.4.2

Rewrite the expression.

$y = x^{1}$

Step 8.3.3.1.2.5

Simplify.

$y = x$

Step 9

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$