Solve (1/m-1-n/m^2-mn):m/n-m=

$$( \frac { 1 } { m - 1 } - \frac { n } { m ^ { 2 } - m n } ) : \frac { m } { n - m } =$$

Evaluate

$$-\frac{m^{2}-2mn+n}{\left(m-1\right)m^{2}}$$

Short Solution Steps

Factor m^{2}-mn.

$$\frac{\frac{1}{m-1}-\frac{n}{m\left(m-n\right)}}{\frac{m}{n-m}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-1 and m\left(m-n\right) is m\left(m-1\right)\left(m-n\right). Multiply \frac{1}{m-1} times \frac{m\left(m-n\right)}{m\left(m-n\right)}. Multiply \frac{n}{m\left(m-n\right)} times \frac{m-1}{m-1}.

$$\frac{\frac{m\left(m-n\right)}{m\left(m-1\right)\left(m-n\right)}-\frac{n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Since \frac{m\left(m-n\right)}{m\left(m-1\right)\left(m-n\right)} and \frac{n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m\left(m-n\right)-n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Do the multiplications in m\left(m-n\right)-n\left(m-1\right).

$$\frac{\frac{m^{2}-mn-mn+n}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Combine like terms in m^{2}-mn-mn+n.

$$\frac{\frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Divide \frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)} by \frac{m}{n-m} by multiplying \frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)} by the reciprocal of \frac{m}{n-m}.

$$\frac{\left(m^{2}-2mn+n\right)\left(n-m\right)}{m\left(m-1\right)\left(m-n\right)m}$$

Extract the negative sign in n-m.

$$\frac{-\left(m-n\right)\left(m^{2}-2mn+n\right)}{mm\left(m-1\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{-\left(m^{2}-2mn+n\right)}{mm\left(m-1\right)}$$

Multiply m and m to get m^{2}.

$$\frac{-\left(m^{2}-2mn+n\right)}{m^{2}\left(m-1\right)}$$

To find the opposite of m^{2}-2mn+n, find the opposite of each term.

$$\frac{-m^{2}+2mn-n}{m^{2}\left(m-1\right)}$$

Use the distributive property to multiply m^{2} by m-1.

$$\frac{-m^{2}+2mn-n}{m^{3}-m^{2}}$$

Expand

$$-\frac{m^{2}-2mn+n}{\left(m-1\right)m^{2}}$$

Short Solution Steps

Factor m^{2}-mn.

$$\frac{\frac{1}{m-1}-\frac{n}{m\left(m-n\right)}}{\frac{m}{n-m}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of m-1 and m\left(m-n\right) is m\left(m-1\right)\left(m-n\right). Multiply \frac{1}{m-1} times \frac{m\left(m-n\right)}{m\left(m-n\right)}. Multiply \frac{n}{m\left(m-n\right)} times \frac{m-1}{m-1}.

$$\frac{\frac{m\left(m-n\right)}{m\left(m-1\right)\left(m-n\right)}-\frac{n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Since \frac{m\left(m-n\right)}{m\left(m-1\right)\left(m-n\right)} and \frac{n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{m\left(m-n\right)-n\left(m-1\right)}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Do the multiplications in m\left(m-n\right)-n\left(m-1\right).

$$\frac{\frac{m^{2}-mn-mn+n}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Combine like terms in m^{2}-mn-mn+n.

$$\frac{\frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)}}{\frac{m}{n-m}}$$

Divide \frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)} by \frac{m}{n-m} by multiplying \frac{m^{2}-2mn+n}{m\left(m-1\right)\left(m-n\right)} by the reciprocal of \frac{m}{n-m}.

$$\frac{\left(m^{2}-2mn+n\right)\left(n-m\right)}{m\left(m-1\right)\left(m-n\right)m}$$

Extract the negative sign in n-m.

$$\frac{-\left(m-n\right)\left(m^{2}-2mn+n\right)}{mm\left(m-1\right)\left(m-n\right)}$$

Cancel out m-n in both numerator and denominator.

$$\frac{-\left(m^{2}-2mn+n\right)}{mm\left(m-1\right)}$$

Multiply m and m to get m^{2}.

$$\frac{-\left(m^{2}-2mn+n\right)}{m^{2}\left(m-1\right)}$$

To find the opposite of m^{2}-2mn+n, find the opposite of each term.

$$\frac{-m^{2}+2mn-n}{m^{2}\left(m-1\right)}$$

Use the distributive property to multiply m^{2} by m-1.

$$\frac{-m^{2}+2mn-n}{m^{3}-m^{2}}$$