Calculus Examples

$x = \frac{y}{\sqrt{y^{2} + 4}} \cdot z$

Step 1

Rewrite the equation as $\frac{y}{\sqrt{y^{2} + 4}} \cdot z = x$ .

$\frac{y}{\sqrt{y^{2} + 4}} \cdot z = x$

Step 2

Simplify $\frac{y}{\sqrt{y^{2} + 4}} \cdot z$ .

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Step 2.1

Multiply $\frac{y}{\sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$\frac{y}{\sqrt{y^{2} + 4}} \cdot \frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}} \cdot z = x$

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

Multiply $\frac{y}{\sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$\frac{y \sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4} \sqrt{y^{2} + 4}} \cdot z = x$

Step 2.2.2

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$\frac{y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1} \sqrt{y^{2} + 4}} \cdot z = x$

Step 2.2.3

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$\frac{y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1} {\sqrt{y^{2} + 4}}^{1}} \cdot z = x$

Step 2.2.4

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

$\frac{y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1 + 1}} \cdot z = x$

Step 2.2.5

Add $1$ and $1$ .

$\frac{y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{2}} \cdot z = x$

Step 2.2.6

Rewrite ${\sqrt{y^{2} + 4}}^{2}$ as $y^{2} + 4$ .

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Step 2.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 4}$ as ${(y^{2} + 4)}^{\frac{1}{2}}$ .

$\frac{y \sqrt{y^{2} + 4}}{{({(y^{2} + 4)}^{\frac{1}{2}})}^{2}} \cdot z = x$

Step 2.2.6.2

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

$\frac{y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{1}{2} \cdot 2}} \cdot z = x$

Step 2.2.6.3

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

$\frac{y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{2}{2}}} \cdot z = x$

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

$\frac{y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{2}{2}}} \cdot z = x$

Step 2.2.6.4.2

Rewrite the expression.

$\frac{y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{1}} \cdot z = x$

Step 2.2.6.5

Simplify.

$\frac{y \sqrt{y^{2} + 4}}{y^{2} + 4} \cdot z = x$

Step 2.3

Combine $\frac{y \sqrt{y^{2} + 4}}{y^{2} + 4}$ and $z$ .

$\frac{y \sqrt{y^{2} + 4} z}{y^{2} + 4} = x$

Step 3

Multiply both sides by $y^{2} + 4$ .

$\frac{y \sqrt{y^{2} + 4} z}{y^{2} + 4} (y^{2} + 4) = x (y^{2} + 4)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $y^{2} + 4$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{y \sqrt{y^{2} + 4} z}{y^{2} + 4} (y^{2} + 4) = x (y^{2} + 4)$

Step 4.1.1.2

Rewrite the expression.

$y \sqrt{y^{2} + 4} z = x (y^{2} + 4)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $x (y^{2} + 4)$ .

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Step 4.2.1.1

Apply the distributive property.

$y \sqrt{y^{2} + 4} z = x y^{2} + x \cdot 4$

Step 4.2.1.2

Move $4$ to the left of $x$ .

$y \sqrt{y^{2} + 4} z = x y^{2} + 4 x$

Step 5

Divide each term in $y \sqrt{y^{2} + 4} z = x y^{2} + 4 x$ by $y \sqrt{y^{2} + 4}$ and simplify.

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Step 5.1

Divide each term in $y \sqrt{y^{2} + 4} z = x y^{2} + 4 x$ by $y \sqrt{y^{2} + 4}$ .

$\frac{y \sqrt{y^{2} + 4} z}{y \sqrt{y^{2} + 4}} = \frac{x y^{2}}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $y$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{y \sqrt{y^{2} + 4} z}{y \sqrt{y^{2} + 4}} = \frac{x y^{2}}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.2.1.2

Rewrite the expression.

$\frac{\sqrt{y^{2} + 4} z}{\sqrt{y^{2} + 4}} = \frac{x y^{2}}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.2.2

Cancel the common factor of $\sqrt{y^{2} + 4}$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{\sqrt{y^{2} + 4} z}{\sqrt{y^{2} + 4}} = \frac{x y^{2}}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.2.2.2

Divide $z$ by $1$ .

$z = \frac{x y^{2}}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 5.3.1.1.1

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

$z = \frac{y (x y)}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.1.2

Cancel the common factors.

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Step 5.3.1.1.2.1

Factor $y$ out of $y \sqrt{y^{2} + 4}$ .

$z = \frac{y (x y)}{y (\sqrt{y^{2} + 4})} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.1.2.2

Cancel the common factor.

$z = \frac{y (x y)}{y \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.1.2.3

Rewrite the expression.

$z = \frac{x y}{\sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.2

Multiply $\frac{x y}{\sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$z = \frac{x y}{\sqrt{y^{2} + 4}} \cdot \frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3

Combine and simplify the denominator.

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Step 5.3.1.3.1

Multiply $\frac{x y}{\sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4} \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.2

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1} \sqrt{y^{2} + 4}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.3

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1} {\sqrt{y^{2} + 4}}^{1}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.4

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

$z = \frac{x y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{1 + 1}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.5

Add $1$ and $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{{\sqrt{y^{2} + 4}}^{2}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6

Rewrite ${\sqrt{y^{2} + 4}}^{2}$ as $y^{2} + 4$ .

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Step 5.3.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 4}$ as ${(y^{2} + 4)}^{\frac{1}{2}}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{{({(y^{2} + 4)}^{\frac{1}{2}})}^{2}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6.2

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

$z = \frac{x y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{1}{2} \cdot 2}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6.3

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

$z = \frac{x y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{2}{2}}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.1.3.6.4.1

Cancel the common factor.

$z = \frac{x y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{\frac{2}{2}}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6.4.2

Rewrite the expression.

$z = \frac{x y \sqrt{y^{2} + 4}}{{(y^{2} + 4)}^{1}} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.3.6.5

Simplify.

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x}{y \sqrt{y^{2} + 4}}$

Step 5.3.1.4

Multiply $\frac{4 x}{y \sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x}{y \sqrt{y^{2} + 4}} \cdot \frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$

Step 5.3.1.5

Combine and simplify the denominator.

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Step 5.3.1.5.1

Multiply $\frac{4 x}{y \sqrt{y^{2} + 4}}$ by $\frac{\sqrt{y^{2} + 4}}{\sqrt{y^{2} + 4}}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y \sqrt{y^{2} + 4} \sqrt{y^{2} + 4}}$

Step 5.3.1.5.2

Move $\sqrt{y^{2} + 4}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y (\sqrt{y^{2} + 4} \sqrt{y^{2} + 4})}$

Step 5.3.1.5.3

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y ({\sqrt{y^{2} + 4}}^{1} \sqrt{y^{2} + 4})}$

Step 5.3.1.5.4

Raise $\sqrt{y^{2} + 4}$ to the power of $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y ({\sqrt{y^{2} + 4}}^{1} {\sqrt{y^{2} + 4}}^{1})}$

Step 5.3.1.5.5

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

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {\sqrt{y^{2} + 4}}^{1 + 1}}$

Step 5.3.1.5.6

Add $1$ and $1$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {\sqrt{y^{2} + 4}}^{2}}$

Step 5.3.1.5.7

Rewrite ${\sqrt{y^{2} + 4}}^{2}$ as $y^{2} + 4$ .

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Step 5.3.1.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 4}$ as ${(y^{2} + 4)}^{\frac{1}{2}}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {({(y^{2} + 4)}^{\frac{1}{2}})}^{2}}$

Step 5.3.1.5.7.2

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

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {(y^{2} + 4)}^{\frac{1}{2} \cdot 2}}$

Step 5.3.1.5.7.3

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

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {(y^{2} + 4)}^{\frac{2}{2}}}$

Step 5.3.1.5.7.4

Cancel the common factor of $2$ .

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Step 5.3.1.5.7.4.1

Cancel the common factor.

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {(y^{2} + 4)}^{\frac{2}{2}}}$

Step 5.3.1.5.7.4.2

Rewrite the expression.

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y {(y^{2} + 4)}^{1}}$

Step 5.3.1.5.7.5

Simplify.

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} + \frac{4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.2

To write $\frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$z = \frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4} \cdot \frac{y}{y} + \frac{4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.3

Write each expression with a common denominator of $(y^{2} + 4) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.1

Multiply $\frac{x y \sqrt{y^{2} + 4}}{y^{2} + 4}$ by $\frac{y}{y}$ .

$z = \frac{x y \sqrt{y^{2} + 4} y}{(y^{2} + 4) y} + \frac{4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.3.2

Reorder the factors of $(y^{2} + 4) y$ .

$z = \frac{x y \sqrt{y^{2} + 4} y}{y (y^{2} + 4)} + \frac{4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.4

Combine the numerators over the common denominator.

$z = \frac{x y \sqrt{y^{2} + 4} y + 4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.5

Simplify the numerator.

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Step 5.3.5.1

Factor $x \sqrt{y^{2} + 4}$ out of $x y \sqrt{y^{2} + 4} y + 4 x \sqrt{y^{2} + 4}$ .

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Step 5.3.5.1.1

Factor $x \sqrt{y^{2} + 4}$ out of $x y \sqrt{y^{2} + 4} y$ .

$z = \frac{x \sqrt{y^{2} + 4} (y \cdot y) + 4 x \sqrt{y^{2} + 4}}{y (y^{2} + 4)}$

Step 5.3.5.1.2

Factor $x \sqrt{y^{2} + 4}$ out of $4 x \sqrt{y^{2} + 4}$ .

$z = \frac{x \sqrt{y^{2} + 4} (y \cdot y) + x \sqrt{y^{2} + 4} (4)}{y (y^{2} + 4)}$

Step 5.3.5.1.3

Factor $x \sqrt{y^{2} + 4}$ out of $x \sqrt{y^{2} + 4} (y \cdot y) + x \sqrt{y^{2} + 4} (4)$ .

$z = \frac{x \sqrt{y^{2} + 4} (y \cdot y + 4)}{y (y^{2} + 4)}$

Step 5.3.5.2

Combine exponents.

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Step 5.3.5.2.1

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

$z = \frac{x \sqrt{y^{2} + 4} (y^{1} y + 4)}{y (y^{2} + 4)}$

Step 5.3.5.2.2

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

$z = \frac{x \sqrt{y^{2} + 4} (y^{1} y^{1} + 4)}{y (y^{2} + 4)}$

Step 5.3.5.2.3

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

$z = \frac{x \sqrt{y^{2} + 4} (y^{1 + 1} + 4)}{y (y^{2} + 4)}$

Step 5.3.5.2.4

Add $1$ and $1$ .

$z = \frac{x \sqrt{y^{2} + 4} (y^{2} + 4)}{y (y^{2} + 4)}$

Step 5.3.6

Cancel the common factor of $y^{2} + 4$ .

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Step 5.3.6.1

Cancel the common factor.

$z = \frac{x \sqrt{y^{2} + 4} (y^{2} + 4)}{y (y^{2} + 4)}$

Step 5.3.6.2

Rewrite the expression.

$z = \frac{x \sqrt{y^{2} + 4}}{y}$