# Simplify ((z^2-w^2)/(z^3-w^3))/((z^2+zw+w^2)/(z^2+2zw+w^2))

z2-w2z3-w3z2+zw+w2z2+2zw+w2
Multiply the numerator by the reciprocal of the denominator.
z2-w2z3-w3⋅z2+2zw+w2z2+zw+w2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=z and b=w.
(z+w)(z-w)z3-w3⋅z2+2zw+w2z2+zw+w2
Since both terms are perfect cubes, factor using the difference of cubes formula, a3-b3=(a-b)(a2+ab+b2) where a=z and b=w.
(z+w)(z-w)(z-w)(z2+zw+w2)⋅z2+2zw+w2z2+zw+w2
Cancel the common factor of z-w.
Cancel the common factor.
(z+w)(z-w)(z-w)(z2+zw+w2)⋅z2+2zw+w2z2+zw+w2
Rewrite the expression.
z+wz2+zw+w2⋅z2+2zw+w2z2+zw+w2
z+wz2+zw+w2⋅z2+2zw+w2z2+zw+w2
Factor using the perfect square rule.
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅z⋅w
Simplify.
2ab=2zw
Factor using the perfect square trinomial rule a2+2ab+b2=(a+b)2, where a=z and b=w.
z+wz2+zw+w2⋅(z+w)2z2+zw+w2
z+wz2+zw+w2⋅(z+w)2z2+zw+w2
Multiply z+wz2+zw+w2⋅(z+w)2z2+zw+w2.
Multiply z+wz2+zw+w2 and (z+w)2z2+zw+w2.
(z+w)(z+w)2(z2+zw+w2)(z2+zw+w2)
Multiply z+w by (z+w)2 by adding the exponents.
Multiply z+w by (z+w)2.
Raise z+w to the power of 1.
(z+w)1(z+w)2(z2+zw+w2)(z2+zw+w2)
Use the power rule aman=am+n to combine exponents.
(z+w)1+2(z2+zw+w2)(z2+zw+w2)
(z+w)1+2(z2+zw+w2)(z2+zw+w2)
(z+w)3(z2+zw+w2)(z2+zw+w2)
(z+w)3(z2+zw+w2)(z2+zw+w2)
Raise z2+zw+w2 to the power of 1.
(z+w)3(z2+zw+w2)1(z2+zw+w2)
Raise z2+zw+w2 to the power of 1.
(z+w)3(z2+zw+w2)1(z2+zw+w2)1
Use the power rule aman=am+n to combine exponents.
(z+w)3(z2+zw+w2)1+1