Simplify (a-5b)/(a^2+ab)*(b^2-a^2)/(10b-2a)

$\frac{a - 5 b}{a^{2} + a b} \cdot \frac{b^{2} - a^{2}}{10 b - 2 a}$

Step 1

Factor $a$ out of $a^{2} + a b$ .

Factor $a$ out of $a^{2}$ .

$\frac{a - 5 b}{a \cdot a + a b} \cdot \frac{b^{2} - a^{2}}{10 b - 2 a}$

Factor $a$ out of $a b$ .

$\frac{a - 5 b}{a \cdot a + a (b)} \cdot \frac{b^{2} - a^{2}}{10 b - 2 a}$

Factor $a$ out of $a \cdot a + a (b)$ .

$\frac{a - 5 b}{a (a + b)} \cdot \frac{b^{2} - a^{2}}{10 b - 2 a}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b$ and $b = a$ .

$\frac{a - 5 b}{a (a + b)} \cdot \frac{(b + a) (b - a)}{10 b - 2 a}$

Step 3

Simplify terms.

Factor $2$ out of $10 b - 2 a$ .

Factor $2$ out of $10 b$ .

$\frac{a - 5 b}{a (a + b)} \cdot \frac{(b + a) (b - a)}{2 (5 b) - 2 a}$

Factor $2$ out of $- 2 a$ .

$\frac{a - 5 b}{a (a + b)} \cdot \frac{(b + a) (b - a)}{2 (5 b) + 2 (- a)}$

Factor $2$ out of $2 (5 b) + 2 (- a)$ .

$\frac{a - 5 b}{a (a + b)} \cdot \frac{(b + a) (b - a)}{2 (5 b - a)}$

Multiply $\frac{a - 5 b}{a (a + b)}$ by $\frac{(b + a) (b - a)}{2 (5 b - a)}$ .

$\frac{(a - 5 b) ((b + a) (b - a))}{a (a + b) (2 (5 b - a))}$

Cancel the common factor of $a - 5 b$ and $5 b - a$ .

Factor $- 1$ out of $a$ .

$\frac{(- 1 (- a) - 5 b) ((b + a) (b - a))}{a (a + b) (2 (5 b - a))}$

Factor $- 1$ out of $- 5 b$ .

$\frac{(- 1 (- a) - (5 b)) ((b + a) (b - a))}{a (a + b) (2 (5 b - a))}$

Factor $- 1$ out of $- 1 (- a) - (5 b)$ .

$\frac{- 1 (- a + 5 b) ((b + a) (b - a))}{a (a + b) (2 (5 b - a))}$

Reorder terms.

$\frac{- 1 (- a + 5 b) ((b + a) (b - a))}{a (a + b) (2 (- a + 5 b))}$

Cancel the common factor.

$\frac{- 1 (- a + 5 b) ((b + a) (b - a))}{a (a + b) (2 (- a + 5 b))}$

Rewrite the expression.

$\frac{- 1 ((b + a) (b - a))}{a (a + b) \cdot (2)}$

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

Reorder terms.

$\frac{- 1 ((a + b) (b - a))}{a (a + b) \cdot (2)}$

Cancel the common factor.

$\frac{- 1 ((a + b) (b - a))}{a (a + b) \cdot 2}$

Rewrite the expression.

$\frac{- 1 (b - a)}{a \cdot (2)}$

Simplify the expression.

Move $2$ to the left of $a$ .

$\frac{- 1 (b - a)}{2 \cdot a}$

Move the negative in front of the fraction.

$- \frac{b - a}{2 a}$

Simplify (a-5b)/(a^2+ab)*(b^2-a^2)/(10b-2a)

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