Question 1151183
<br>
{{{0 = x^5-(x^3)(y^3)-12393}}}<br>
Rewrite the equation as<br>
{{{x^3(x^2-y^3) = 12393}}}<br>
Then find the prime factorization of 12393:<br>
{{{12393 = (3^6)(17)}}}<br>
So<br>
{{{(x^3)(x^2-y^3) = (3^6)(17)}}}<br>
Since x and y are positive integers,
{{{x^3= 3^6}}}
means either {{{x=3}}} or {{{x=3^2=9}}}.<br>
If x=3,
{{{(x^3)(x^2-y^3) = (27)(9-y^3) = 12393}}}
which is clearly not possible.<br>
If x=9,
{{{(x^3)(x^2-y^3) = (729)(81-y^3) = 729*17}}}
{{{81-y^3=17}}}
{{{y^3=64}}}
{{{y=4}}}<br>
So x=9 and y=4; and then<br>
ANSWER: {{{sqrt(y/x) = sqrt(4/9) = 2/3}}}<br>