Algebra - Challenging Question! Can you solve it?

[@yurenchu]

Solve for integers m and n , given
³√( 6√45 - 17 ) = m/(√n + 1)
Lefthandside:
³√( 6√45 - 17 ) =
= ³√( (48√45 - 136) / 8 )
= ³√( (45√45 + 3√45 - 135 - 1) / 2³ )
= ³√( (45√45 - 3*45 + 3√45 - 1) / 2³ )
= ³√( (√45 - 1)³ / 2³ )
= ³√( [ (√45 - 1)/2 ]³ )
= (√45 - 1)/2
Righthandside:
m/(√n + 1) =
... multiply numerator and denominator by (√n - 1) ...
= [ m*(√n - 1) ] / [ (√n + 1)*(√n - 1) ]
= [ m*(√n - 1) ] / (n - 1)
= (√n - 1) * m/(n-1)
Equating LHS and RHS:
(√45 - 1)/2 = (√n - 1) * m/(n-1)
(√45 - 1) * 1/2 = (√n - 1) * m/(n-1)
... set n = 45 ...
1/2 = m/(45 - 1)
1/2 = m/44
m = 44*(1/2) = 22
==> n = 45 , m = 22

[@Mathsandolympiad]

Thanks for watching