Simplify (((a+5)(a-2))/(a^2-a))÷((3(a+5))/((a-1)(a-2)))

$\frac{(a + 5) (a - 2)}{a^{2} - a} \div \frac{3 (a + 5)}{(a - 1) (a - 2)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{(a + 5) (a - 2)}{a^{2} - a} \cdot \frac{(a - 1) (a - 2)}{3 (a + 5)}$

Step 2

Cancel the common factor of $a + 5$ .

Factor $a + 5$ out of $3 (a + 5)$ .

$\frac{(a + 5) (a - 2)}{a^{2} - a} \cdot \frac{(a - 1) (a - 2)}{(a + 5) \cdot 3}$

Cancel the common factor.

$\frac{(a + 5) (a - 2)}{a^{2} - a} \cdot \frac{(a - 1) (a - 2)}{(a + 5) \cdot 3}$

Rewrite the expression.

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

Step 3

Multiply $\frac{a - 2}{a^{2} - a}$ by $\frac{(a - 1) (a - 2)}{3}$ .

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

Step 4

Raise $a - 2$ to the power of $1$ .

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

Step 5

Raise $a - 2$ to the power of $1$ .

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

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

Factor $a$ out of $- a$ .

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

Factor $a$ out of $a \cdot a + a \cdot - 1$ .

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

Step 9

Cancel the common factor of $a - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 10

Move $3$ to the left of $a$ .

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