Question

Simplify each rational expression.

3a(a-4)^2 over a^2(a-4) I don't understand how the answer would be 3(a-4) over a because the a-4 would cancel out???

4-y^2 over y^2-2y once I simplify I get (2-y)(2+y) over (y-2)(y+2) I don't understand how the answer is -(y+2) over y

9-r^2 over r^2-3 once I simplify i get (3-r)(3+r) over (r-3)(r+3) I don't understand how the answer is -(r+3) over r

Answer 1

3a(a-4)^2 over a^2(a-4)

3a(a-4)(a-4) over a*a*(a-4)
You are correct that an (a-4) would cancel, but you are left with another one because the top had the power of two on it. In addition, 1 a would also cancel, leaving you with 3(a-4)/a

4-y^2 over y^2-2y
Rewrite the numerator in standard form
-y^2+4 and then factor out a -1, leaving you with -1(y^2 - 4) which then factors into: -1(y+2)(y-2)
Then the denominator factors into:
y(y-2). As a result, the (y-2) factors cancel, leaving you with -1(y+2)/y

9-r^2 over r^2-3
(Is the denominator supposed to be r^2-3r?)
Again, rewrite: -r^2+9 --> -1(r^2-9) --> then as -1(r+3)(r-3)
Denominator: r(r-3)
Cancel out the (r-3)'s and you are left with -(r+3)/r