Basic Math Examples

$\frac{\frac{z}{z + 4}}{\frac{1}{z} + \frac{1}{z + 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{z}{z + 4} \cdot \frac{1}{\frac{1}{z} + \frac{1}{z + 4}}$

Step 2

Simplify the denominator.

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Step 2.1

To write $\frac{1}{z}$ as a fraction with a common denominator, multiply by $\frac{z + 4}{z + 4}$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{1}{z} \cdot \frac{z + 4}{z + 4} + \frac{1}{z + 4}}$

Step 2.2

To write $\frac{1}{z + 4}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{1}{z} \cdot \frac{z + 4}{z + 4} + \frac{1}{z + 4} \cdot \frac{z}{z}}$

Step 2.3

Write each expression with a common denominator of $z (z + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{z}$ by $\frac{z + 4}{z + 4}$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{z + 4}{z (z + 4)} + \frac{1}{z + 4} \cdot \frac{z}{z}}$

Step 2.3.2

Multiply $\frac{1}{z + 4}$ by $\frac{z}{z}$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{z + 4}{z (z + 4)} + \frac{z}{(z + 4) z}}$

Step 2.3.3

Reorder the factors of $(z + 4) z$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{z + 4}{z (z + 4)} + \frac{z}{z (z + 4)}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{z}{z + 4} \cdot \frac{1}{\frac{z + 4 + z}{z (z + 4)}}$

Step 2.5

Reorder terms.

$\frac{z}{z + 4} \cdot \frac{1}{\frac{z + z + 4}{z (z + 4)}}$

Step 2.6

Rewrite $\frac{z + z + 4}{z (z + 4)}$ in a factored form.

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Step 2.6.1

Add $z$ and $z$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{2 z + 4}{z (z + 4)}}$

Step 2.6.2

Factor $2$ out of $2 z + 4$ .

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Step 2.6.2.1

Factor $2$ out of $2 z$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{2 (z) + 4}{z (z + 4)}}$

Step 2.6.2.2

Factor $2$ out of $4$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{2 z + 2 \cdot 2}{z (z + 4)}}$

Step 2.6.2.3

Factor $2$ out of $2 z + 2 \cdot 2$ .

$\frac{z}{z + 4} \cdot \frac{1}{\frac{2 (z + 2)}{z (z + 4)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{z}{z + 4} (1 \frac{z (z + 4)}{2 (z + 2)})$

Step 4

Multiply $\frac{z (z + 4)}{2 (z + 2)}$ by $1$ .

$\frac{z}{z + 4} \cdot \frac{z (z + 4)}{2 (z + 2)}$

Step 5

Cancel the common factor of $z + 4$ .

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Step 5.1

Factor $z + 4$ out of $z (z + 4)$ .

$\frac{z}{z + 4} \cdot \frac{(z + 4) z}{2 (z + 2)}$

Step 5.2

Cancel the common factor.

$\frac{z}{z + 4} \cdot \frac{(z + 4) z}{2 (z + 2)}$

Step 5.3

Rewrite the expression.

$z \frac{z}{2 (z + 2)}$

Step 6

Combine $z$ and $\frac{z}{2 (z + 2)}$ .

$\frac{z \cdot z}{2 (z + 2)}$

Step 7

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

$\frac{z^{1} z}{2 (z + 2)}$

Step 8

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

$\frac{z^{1} z^{1}}{2 (z + 2)}$

Step 9

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

$\frac{z^{1 + 1}}{2 (z + 2)}$

Step 10

Add $1$ and $1$ .

$\frac{z^{2}}{2 (z + 2)}$