We take m = 2, 3, 4 etc.
We see that
When m = 2 :
m2 = 22 = 2 x 2 = 4 and m3 = 23 = 2 x 2 x 2 = 8
Clearly, 4 < 8, i.e. m2 < m3
When m = 3 :
m2 = 32 = 3 x 3 = 9 and m3 = 33 = 3 x 3 x 3 = 27
Clearly, 9 < 27, i.e. m2 < m3
When m = 4 :
m2 = 42 = 4 x 4 = 16 and m3 = 43 = 4 x 4 x 4 = 64
Clearly, 16 < 64, i.e. m2 < m3
But when m = 1,
m2 = 12 = 1 x 1 = 1 and m3 = 13 = 1 x 1 x 1 = 1
Then m2 < m3
Thus we can say that for any positive integer (natural number) m > 1, m2 < m3 is true.
Now, consider m = – 1, – 2, – 3 etc.
When m = – 1 :
m2 = (- 1)2 = (- 1) x (- 1) = 1 and m3 = (- 1)3 = (- 1) x (- 1) x (- 1) = -1
Clearly, 1 > – 1, i.e. m2 > m3
When m = -2 :
m2 = (- 2)2 = (- 2) x (- 2) = 4 and m3 = (- 2)3 = (- 2) x (- 2) x (- 2) = – 8
Clearly, 4 > – 8, i.e. m2 > m3
Thus we can say that for any negative integer m, m2 < m3 is false.