a becomes 8 times larger when b is doubled.
Let's take the original expression 3b^3/c and effectively double b by multiplying by 2. Then we will take the new expression and divide by the old expression and see what happens.
a = 3b^3/c
a' = 3(2b)^3/c
a'/a = 3(2b)^3/c / 3b^3/c
You can divide by a fraction by swapping the top and bottom and then multiplying. So
a'/a= 3(2b)^3/c * c/3b^3
The c terms cancel, giving:
a'/a= 3(2b)^3/1 * 1/3b^3
The 3's cancel, giving:
a'/a= (2b)^3/1 * 1/b^3
Expand the parenthesis
a'/a= 8b^3/1 * 1/b^3
Cancel the b^3 terms
a'/a= 8/1 * 1/1 = 8 * 1 = 8
So a' is 8 times larger than a.
