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The question states that the Transformation T(p', q') is of the same form as the Transformation T(p, q), and that the way they're related is T^2n = T'^n.

Applying transformation T(p, q) twice on (a, b), we get:
a = bq + aq + ap                 - (1)
b = bp + aq

a = (bp + aq)q + (bq + aq + ap)q + (bq + aq + ap)p
b = (bp + aq)p + (bq + aq + ap)q

Solving these, so we get the above equations in form (1), we get,

a = b(q^2 + 2pq) + a(q^2 + 2pq) + a(p^2 + q^2)
b = b(p^2 + q^2) + a(q^2 + 2pq)

Therefore p' = p^2 + q^2
      and q' = q^2 + 2pq