How do you simplify #(2a^2+ab-b^2)/(a^3+b^3) ÷ (2a^2b^2-ab^3)/(2a^2-2ab+2b^2)#?

1 Answer
Mar 3, 2017

Answer:

#=2/(ab^2)#

Explanation:

The first step in algebraic fractions is to factorise wherever possible.

#color(red)((2a^2+ab-b^2))/color(green)((a^3+b^3)) ÷ color(blue)((2a^2b^2-ab^3))/color(purple)((2a^2-2ab+2b^2))#

#color(red)("quadratic trinomial")/color(green)("sum of cubes") ÷ color(blue)("common factor")/color(purple)("common factor")#

#=color(red)((2a-b)(a+b))/color(green)((a+b)(a^2-ab+b^2)) ÷ color(blue)(ab^2(2a -b))/color(purple)(2(a^2-ab+b^2))#

To divide: multiply by the reciprocal

#=color(red)((2a-b)(a+b))/color(green)((a+b)(a^2-ab+b^2)) xx color(purple)(2(a^2-ab+b^2))/color(blue)(ab^2(2a -b))#

Cancel like factors

#=(cancel((2a-b))cancel((a+b)))/(cancel((a+b))cancel((a^2-ab+b^2))) xx color(purple)(2cancel((a^2-ab+b^2)))/(ab^2cancel((2a -b)))#

#=2/(ab^2)#