Which Expression will I Factorise?!

Algebra Level 3

$\begin{cases} x^{2}+y^{2}=z^{2} \\ xy=z \\ (x+y)^{2} = 80 \end{cases}$

If the above system of equations holds true for some positive real numbers $x$ , $y$ , and $z$ , then find

$\large \frac{z^{4}}{\left(\frac{x^3y+xy^3}{z^3}\right)} .$

Hint: Use factorization.

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1 solution

Fidel Simanjuntak
Jul 18, 2016

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

$80=z^{2}+2z$

$z^{2}+2z-80=0$

$(z+10)(z-8)=0$

$z$ is a positive real number, so $z=8$ .

Now, note that

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

$=\frac{(z)(z^2)}{z^3}=1$

So,

$\frac{z^{4}}{(\frac{x^3y+xy^3}{z^3})} = z^{4}$

$=8^{4} = \boxed{4096}$