Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
$\frac{a+b}{a^{2/3}-a^{1/3}{b}^{1/3}+b^{2/3}}-\frac{a-b}{a^{2/3}+a^{1/3}{b}^{1/3}{b}^{2/3}}-\frac{a^{2/3}-b^{2/3}}{a^{1/3}-b^{1/3}}\cdot$

Answer 1

This question is wrong

Correct question is,

$\frac{a+b}{a^{2/3}-a^{1/3}.b^{1/3}+b^{2/3}}-\frac{a-b}{a^{2/3}+a^{1/3}.b^{1/3}+b^{2/3}}-\frac{a^{2/3}-b^{2/3}}{a^{1/3}-b^{1/3}}\cdot$

here put $a^{1/3}=x,$ and $b^{1/3}=y$

then, we get,

$\frac{x^3+y^3}{x^2-xy+y^2}-\frac{x^3-y^3}{x^2+xy+y^2}-\frac{x^2-y^2}{x-y}$

$=\frac{(x+y)(x^2-xy+y^2)}{(x^2-xy+y^2)}-\frac{(x-y)(x^2+xy+y^2)}{x^2+xy+y^2}-\frac{(x-y)(x+y)}{(x-y)}$

$=(x+y)-(x-y)(x+y)$

$=x+y-x+y-x-y$

$=-(x-y)$

$=y-x$

$=b^{1/3}-a^{1/3}$