>>> def definitely_too_big(x):
...     too_big = 0
...     while 9 * len([int(digit) for digit in str(too_big **x)]) >= too_big:
...         too_big += 1
...     
...     return(too_big)
... 
>>> unicode_exponents = ['⁰', '¹', '²', '³', '⁴', '⁵', '⁶', '⁷', '⁸', '⁹']
>>> for exp in range(10):
...     for num in range(definitely_too_big(exp)):
...         if sum(int(digit) for digit in str(num ** exp)) == num:
...             print('({0}){exp} = {1} = {2}{exp}'.format(' + '.join(digit for digit in str(num ** exp)), num ** exp, num, exp=unicode_exponents[exp]))
...     print()
... 
(1)⁰ = 1 = 1⁰

(0)¹ = 0 = 0¹
(1)¹ = 1 = 1¹
(2)¹ = 2 = 2¹
(3)¹ = 3 = 3¹
(4)¹ = 4 = 4¹
(5)¹ = 5 = 5¹
(6)¹ = 6 = 6¹
(7)¹ = 7 = 7¹
(8)¹ = 8 = 8¹
(9)¹ = 9 = 9¹

(0)² = 0 = 0²
(1)² = 1 = 1²
(8 + 1)² = 81 = 9²

(0)³ = 0 = 0³
(1)³ = 1 = 1³
(5 + 1 + 2)³ = 512 = 8³
(4 + 9 + 1 + 3)³ = 4913 = 17³
(5 + 8 + 3 + 2)³ = 5832 = 18³
(1 + 7 + 5 + 7 + 6)³ = 17576 = 26³
(1 + 9 + 6 + 8 + 3)³ = 19683 = 27³

(0)⁴ = 0 = 0⁴
(1)⁴ = 1 = 1⁴
(2 + 4 + 0 + 1)⁴ = 2401 = 7⁴
(2 + 3 + 4 + 2 + 5 + 6)⁴ = 234256 = 22⁴
(3 + 9 + 0 + 6 + 2 + 5)⁴ = 390625 = 25⁴
(6 + 1 + 4 + 6 + 5 + 6)⁴ = 614656 = 28⁴
(1 + 6 + 7 + 9 + 6 + 1 + 6)⁴ = 1679616 = 36⁴

(0)⁵ = 0 = 0⁵
(1)⁵ = 1 = 1⁵
(1 + 7 + 2 + 1 + 0 + 3 + 6 + 8)⁵ = 17210368 = 28⁵
(5 + 2 + 5 + 2 + 1 + 8 + 7 + 5)⁵ = 52521875 = 35⁵
(6 + 0 + 4 + 6 + 6 + 1 + 7 + 6)⁵ = 60466176 = 36⁵
(2 + 0 + 5 + 9 + 6 + 2 + 9 + 7 + 6)⁵ = 205962976 = 46⁵

(0)⁶ = 0 = 0⁶
(1)⁶ = 1 = 1⁶
(3 + 4 + 0 + 1 + 2 + 2 + 2 + 4)⁶ = 34012224 = 18⁶
(8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5)⁶ = 8303765625 = 45⁶
(2 + 4 + 7 + 9 + 4 + 9 + 1 + 1 + 2 + 9 + 6)⁶ = 24794911296 = 54⁶
(6 + 8 + 7 + 1 + 9 + 4 + 7 + 6 + 7 + 3 + 6)⁶ = 68719476736 = 64⁶

(0)⁷ = 0 = 0⁷
(1)⁷ = 1 = 1⁷
(6 + 1 + 2 + 2 + 2 + 0 + 0 + 3 + 2)⁷ = 612220032 = 18⁷
(1 + 0 + 4 + 6 + 0 + 3 + 5 + 3 + 2 + 0 + 3)⁷ = 10460353203 = 27⁷
(2 + 7 + 5 + 1 + 2 + 6 + 1 + 4 + 1 + 1 + 1)⁷ = 27512614111 = 31⁷
(5 + 2 + 5 + 2 + 3 + 3 + 5 + 0 + 1 + 4 + 4)⁷ = 52523350144 = 34⁷
(2 + 7 + 1 + 8 + 1 + 8 + 6 + 1 + 1 + 1 + 0 + 7)⁷ = 271818611107 = 43⁷
(1 + 1 + 7 + 4 + 7 + 1 + 1 + 1 + 3 + 9 + 8 + 3 + 7)⁷ = 1174711139837 = 53⁷
(2 + 2 + 0 + 7 + 9 + 8 + 4 + 1 + 6 + 7 + 5 + 5 + 2)⁷ = 2207984167552 = 58⁷
(6 + 7 + 2 + 2 + 9 + 8 + 8 + 8 + 1 + 8 + 4 + 3 + 2)⁷ = 6722988818432 = 68⁷

(0)⁸ = 0 = 0⁸
(1)⁸ = 1 = 1⁸
(2 + 0 + 0 + 4 + 7 + 6 + 1 + 2 + 2 + 3 + 1 + 9 + 3 + 6)⁸ = 20047612231936 = 46⁸
(7 + 2 + 3 + 0 + 1 + 9 + 6 + 1 + 3 + 3 + 9 + 1 + 3 + 6)⁸ = 72301961339136 = 54⁸
(2 + 4 + 8 + 1 + 5 + 5 + 7 + 8 + 0 + 2 + 6 + 7 + 5 + 2 + 1)⁸ = 248155780267521 = 63⁸

(0)⁹ = 0 = 0⁹
(1)⁹ = 1 = 1⁹
(3 + 9 + 0 + 4 + 3 + 0 + 5 + 9 + 1 + 2 + 3 + 1 + 3 + 3 + 4 + 4)⁹ = 3904305912313344 = 54⁹
(4 + 5 + 8 + 4 + 8 + 5 + 0 + 0 + 7 + 1 + 8 + 4 + 4 + 9 + 0 + 3 + 1)⁹ = 45848500718449031 = 71⁹
(1 + 5 + 0 + 0 + 9 + 4 + 6 + 3 + 5 + 2 + 9 + 6 + 9 + 9 + 9 + 1 + 2 + 1)⁹ = 150094635296999121 = 81⁹