Question

Solve $\frac{x^{-3}-y^{-3}}{x^{-3}{y}^{-1}+(xy{)}^{-2}+y^{-3}{x}^{-1}}$

• A. $x+y$
• B. $y-x$
• C. $\frac{1}{x}-\frac{1}{y}$
• D. $\frac{1}{x}+\frac{1}{y}$

Answer 1

Answer: (B)

$y-x$

$\frac{x^{-3}-y^{-3}}{x^{-3}{y}^{-1}+(xy{)}^{-2}+y^{-3}{x}^{-1}}$

Multiplying the denominator as well as the numerator by $x^3{y}^3.$

$=\frac{x^3{y}^3{x}^{-3}-x^3{y}^3{y}^{-3}}{x^3{y}^3{x}^{-3}{y}^{-1}+x^3{y}^3(xy{)}^{-2}+x^3{y}^3{y}^{-3}{x}^{-1}}$
$=\frac{y^3-x^3}{y^2+xy+x^2}$

Since we can write $y^3-x^3$ as $(y-x)\times (x^2+xy+y^2)$,

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

the only term that remains in the numerator is $y-x.$

$\frac{x^{-3}-y^{-3}}{x^{-3}{y}^{-1}+(xy{)}^{-2}+y^{-3}{x}^{-1}}=y-x$