When a quantity is required to be multiplied by its equal, the product is called the square of the quantity. Thus a x a = a* (Art. 412). If the square be multiplied by the original quantity the result is a cube; or, a2 x a = a3; or, generally, for -

a, a a, a a a, aaaa, aaaaa, we put -

a, a2, a2, a4, a5;

in which the small number at the upper right-hand corner indicates the number of times the quantity occurs in the expression. Thus if a - 2, then a2=2x2=4, a3 = 4 x 2 = 8, a4 = 8 x 2 = 16, a5 = 16 x 2 = 32; any term in the series of powers may be found by multiplying the preceding one by a, or by dividing the succeeding one by a.

Thus a4x a = a5and a/a5 =a4.