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As usual with those equation involving exponentials :

sage: solve(k == 50*k^(3/5),k, to_poly_solve=True)
[k == 0, k == 12500*sqrt(2)]
sage: solve(k == 50*k^(3/5),k, algorithm="sympy")
[k == 0, k == 12500*sqrt(2)]
sage: (k==50*k^(3/5)).log().log_expand().solve(k)
[k == 12500*sqrt(2)]

HTH,