Algebra Examples

$\frac{m^{- 1} + z^{- 1}}{m^{- 1} - z^{- 1}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{m} + z^{- 1}}{m^{- 1} - z^{- 1}}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{m} + \frac{1}{z}}{m^{- 1} - z^{- 1}}$

Step 1.3

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

$\frac{\frac{1}{m} \cdot \frac{z}{z} + \frac{1}{z}}{m^{- 1} - z^{- 1}}$

Step 1.4

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

$\frac{\frac{1}{m} \cdot \frac{z}{z} + \frac{1}{z} \cdot \frac{m}{m}}{m^{- 1} - z^{- 1}}$

Step 1.5

Write each expression with a common denominator of $m z$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{1}{m}$ by $\frac{z}{z}$ .

$\frac{\frac{z}{m z} + \frac{1}{z} \cdot \frac{m}{m}}{m^{- 1} - z^{- 1}}$

Step 1.5.2

Multiply $\frac{1}{z}$ by $\frac{m}{m}$ .

$\frac{\frac{z}{m z} + \frac{m}{z m}}{m^{- 1} - z^{- 1}}$

Step 1.5.3

Reorder the factors of $z m$ .

$\frac{\frac{z}{m z} + \frac{m}{m z}}{m^{- 1} - z^{- 1}}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{\frac{z + m}{m z}}{m^{- 1} - z^{- 1}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{z + m}{m z}}{\frac{1}{m} - z^{- 1}}$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{z + m}{m z}}{\frac{1}{m} - \frac{1}{z}}$

Step 2.3

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

$\frac{\frac{z + m}{m z}}{\frac{1}{m} \cdot \frac{z}{z} - \frac{1}{z}}$

Step 2.4

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

$\frac{\frac{z + m}{m z}}{\frac{1}{m} \cdot \frac{z}{z} - \frac{1}{z} \cdot \frac{m}{m}}$

Step 2.5

Write each expression with a common denominator of $m z$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{1}{m}$ by $\frac{z}{z}$ .

$\frac{\frac{z + m}{m z}}{\frac{z}{m z} - \frac{1}{z} \cdot \frac{m}{m}}$

Step 2.5.2

Multiply $\frac{1}{z}$ by $\frac{m}{m}$ .

$\frac{\frac{z + m}{m z}}{\frac{z}{m z} - \frac{m}{z m}}$

Step 2.5.3

Reorder the factors of $z m$ .

$\frac{\frac{z + m}{m z}}{\frac{z}{m z} - \frac{m}{m z}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{\frac{z + m}{m z}}{\frac{z - m}{m z}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{z + m}{m z} \cdot \frac{m z}{z - m}$

Step 4

Cancel the common factor of $m z$ .

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Step 4.1

Cancel the common factor.

$\frac{z + m}{m z} \cdot \frac{m z}{z - m}$

Step 4.2

Rewrite the expression.

$(z + m) \frac{1}{z - m}$

Step 5

Multiply $z + m$ by $\frac{1}{z - m}$ .

$\frac{z + m}{z - m}$