Simplify (ma-mb)/(3n^2)*(2m)/(nb-na)

$\frac{m a - m b}{3 n^{2}} \cdot \frac{2 m}{n b - n a}$

Step 1

Simplify the numerator.

Factor $m$ out of $m a - m b$ .

Factor $m$ out of $m a$ .

$\frac{m (a) - m b}{3 n^{2}} \cdot \frac{2 m}{n b - n a}$

Factor $m$ out of $- m b$ .

$\frac{m (a) + m (- 1 b)}{3 n^{2}} \cdot \frac{2 m}{n b - n a}$

Factor $m$ out of $m (a) + m (- 1 b)$ .

$\frac{m (a - 1 b)}{3 n^{2}} \cdot \frac{2 m}{n b - n a}$

Rewrite $- 1 b$ as $- b$ .

$\frac{m (a - b)}{3 n^{2}} \cdot \frac{2 m}{n b - n a}$

Step 2

Simplify the denominator.

Factor $n$ out of $n b - n a$ .

Factor $n$ out of $n b$ .

$\frac{m (a - b)}{3 n^{2}} \cdot \frac{2 m}{n (b) - n a}$

Factor $n$ out of $- n a$ .

$\frac{m (a - b)}{3 n^{2}} \cdot \frac{2 m}{n (b) + n (- 1 a)}$

Factor $n$ out of $n (b) + n (- 1 a)$ .

$\frac{m (a - b)}{3 n^{2}} \cdot \frac{2 m}{n (b - 1 a)}$

Rewrite $- 1 a$ as $- a$ .

$\frac{m (a - b)}{3 n^{2}} \cdot \frac{2 m}{n (b - a)}$

Step 3

Combine.

$\frac{m (a - b) (2 m)}{3 n^{2} (n (b - a))}$

Step 4

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$\frac{m \cdot m (a - b) \cdot 2}{3 n^{2} (n (b - a))}$

Multiply $m$ by $m$ .

$\frac{m^{2} (a - b) \cdot 2}{3 n^{2} (n (b - a))}$

Step 5

Multiply $n^{2}$ by $n$ by adding the exponents.

Move $n$ .

$\frac{m^{2} (a - b) \cdot 2}{3 (n \cdot n^{2}) (b - a)}$

Multiply $n$ by $n^{2}$ .

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

$\frac{m^{2} (a - b) \cdot 2}{3 (n^{1} n^{2}) (b - a)}$

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

$\frac{m^{2} (a - b) \cdot 2}{3 n^{1 + 2} (b - a)}$

Add $1$ and $2$ .

$\frac{m^{2} (a - b) \cdot 2}{3 n^{3} (b - a)}$

Step 6

Cancel the common factor of $a - b$ and $b - a$ .

Factor $- 1$ out of $a$ .

$\frac{m^{2} (- 1 (- a) - b) \cdot 2}{3 n^{3} (b - a)}$

Factor $- 1$ out of $- b$ .

$\frac{m^{2} (- 1 (- a) - (b)) \cdot 2}{3 n^{3} (b - a)}$

Factor $- 1$ out of $- 1 (- a) - (b)$ .

$\frac{m^{2} (- 1 (- a + b)) \cdot 2}{3 n^{3} (b - a)}$

Reorder terms.

$\frac{m^{2} (- 1 (- a + b)) \cdot 2}{3 n^{3} (- a + b)}$

Cancel the common factor.

$\frac{m^{2} (- 1 (- a + b)) \cdot 2}{3 n^{3} (- a + b)}$

Rewrite the expression.

$\frac{m^{2} \cdot (- 1) \cdot 2}{3 n^{3}}$

Step 7

Simplify the expression.

Multiply $2$ by $- 1$ .

$\frac{m^{2} \cdot - 2}{3 n^{3}}$

Move $- 2$ to the left of $m^{2}$ .

$\frac{- 2 \cdot m^{2}}{3 n^{3}}$

Move the negative in front of the fraction.

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

Simplify (ma-mb)/(3n^2)*(2m)/(nb-na)

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