Expert: Abe Mantell - 7/7/2011

Question
8.   Simplify: x^4-2x^2y^2+y^4/x^4-x^3y-xy^3+y^4

10. Simplify: z(z-1)= z(z+1)

22. Simplify: 3x^2-8x+4 / 9x^2-4 ÷ 3x^2-5x-2/ 9x^2-3x-2

23. Simplify: ( It's a fraction inside of a fraction)
       x^2-y^2
      ___________
          x +y
       x^4-y^4

a) x^2+y^2     b) 1    c) 1/ (x^2+y^2) ( x+1)   d) x^4-y^4

27. Simplify: (x+3/ 5) - ( 2x+1/10)

28.  SImplify: (x+3/5) + (x-2/2x)

32. Simplify: 1/p^2 - 1/q^2
                      ______________
                       2/p^2 - 1/pq - 1/q^2

33. Simplify: x^-2 - x^2/ x^-1-x

34. Simplify: 1/x - 1/y
                      __________
                     y/x- x/y

Answer
Hello Victoria,

You have many questions here.  I'll just answer the first few.  Try the
others, if you need help, let m eknow.

8.  x^4-2x^2y^2+y^4/x^4-x^3y-xy^3+y^4
I gather you mean:  (x^4-2x^2y^2+y^4)/(x^4-x^3y-xy^3+y^4)...yes?
If so, then the numerator factors to: (x^2-y^2)^2 = [(x+y)(x-y)]^2
and the denominator is: (x^4-x^3 y)-(xy^3-y^4)=x^3(x-y)-y^3(x-y)=(x^3-y^3)(x-y)
but x^3-y^3 factors to (x-y)(x^2+xy+y^2)

So, (x^4-2x^2y^2+y^4)/(x^4-x^3y-xy^3+y^4)
= [(x+y)^2 (x-y)^2]/[(x-y)^2 (x^2+xy+y^2)]
= (x+y)^2/(x^2+xy+y^2)

10. Simplify or solve? z(z-1)= z(z+1)
z(z-1)-z(z+1)=0
z[(z-1)-(z+1)]=0
z[-2]=0 ==> z=0

23. Simplify: ( It's a fraction inside of a fraction)
      x^2-y^2
     __________
        x+y
   --------------
      x^4-y^4

     (x+y)(x-y)
=     __________
        x +y
 -----------------
 (x^2-y^2)(x^2+y^2)

    (x+y)(x-y)
=    __________
       x+y
 -------------------
 (x-y)(x+y)(x^2+y^2)

          x-y
=  ___________________
  (x-y)(x+y)(x^2+y^2)

          1
=   ________________   answer (c)
    (x+y)(x^2+y^2)

a) x^2+y^2     b) 1    c) 1/ (x^2+y^2) ( x+1)   d) x^4-y^4

OK?

TTYL, Abe