How do you simplify #(x ^ 4 - 16y ^ 4)/( ( x² + 4y² ) ( x-2y ))#?

Question

How do you simplify #(x ^ 4  -  16y ^ 4)/( ( x² + 4y² ) ( x-2y ))#?

Ethan Adkins · Accepted Answer

x + 2y

Begin by factorising the numerator, which is a #color(blue)"difference of squares"# and in general factorises as #color(red)(|bar(ul(color(white)(a/a)color(black)(a^2-b^2=(a-b)(a+b))color(white)(a/a)|))) ........ (A)# #x^4=(x^2)^2" and " 16y^4=(4y^2)^2# #rArra=x^2" and " b=4y^2# Substitute a and b into (A) #rArrx^4-16y^4=(x^2-4y^2)(x^2+4y^2)# #rArr((x^2-4y^2)cancel((x^2+4y^2)))/((cancel((x^2+4y^2))(x-2y)# Now #x^2-4y^2" is also a " color(blue)"difference of squares"# #x^2=(x)^2" and " 4y^2=(2y)^2rArra=x" and " b=2y# substitute a and b into (A) #x^2-4y^2=(x-2y)(x+2y)# #rArr(cancel((x-2y))(x+2y))/(cancel((x-2y)))=x+2y#

Eva Abramson · Answer

undefined To simplify the expression (x^4 - 16y^4)/((x^2 + 4y^2)(x-2y)), we can factor the numerator and denominator. The numerator can be factored as a difference of squares: (x^2 + 4y^2)(x^2 - 4y^2). The denominator can be factored as a difference of squares as well: (x^2 + 4y^2)(x - 2y).

Now, we can cancel out the common factors of (x^2 + 4y^2) from the numerator and denominator, leaving us with (x^2 - 4y^2)/(x - 2y).

Further simplifying, we can factor the numerator as a difference of squares: (x + 2y)(x - 2y)/(x - 2y).

Finally, we can cancel out the common factor of (x - 2y) from the numerator and denominator, resulting in the simplified expression of x + 2y.