Don't use quadratic equation

Algebra Level 3

If $x-y=8$ and $x^2+y^2=80$ ,then $x^{-3}-y^{-3}=z$ .

If $|z|=\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

1 solution

Tommy Li
May 23, 2016

$x - y = 8$

$x^2 - 2xy + y^2 = 64$

$80 - 2xy = 64$

$xy = 8$

$x^{-3} - y^{-3}$

$=\frac{1}{x^3} - \frac{1}{y^3}$

$=\frac{y^3-x^3}{x^3y^3}$

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

$=\frac{(-8)(80+8)}{8^3}$

$=-\frac{11}{8}$

$\lvert -\frac{11}{8} \rvert= \frac{11}{8}$

$a+b=11+8=19$

I should say the fraction is the simplest form....

Tommy Li - 5 years ago

Good use of algebraic manipulation to get the desired result. I have edited the Latex so that it is easier to read your solution.

Pranshu Gaba - 5 years ago

Thanks you so much

Tommy Li - 5 years ago