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Consider the number 4 x 4 x 4 x 4 x 4 We can write this in the abbreviated form of 45. Just as 4+4+4+4+4 can be written as 5 x 4, so 4 x 4 x 4 x 4 x 4 can be written as 45. 45 is called a power. It is the fifth power of 4. It is usually read as "four to the fifth". For the power ap, a is called the base and p the exponent. So 4 is the base and 5 is the exponent of the power 45. The exponent is often loosely referred to as the power.
Rules for calculating with powers: (Note that these are all derivable from the definition of a power.)
am x an = am+n
From the definition of am and an we can see that
am x an = (a x a x …..x a)[m times] x (a x a x …..x a)[n times]
= (a x a x a x a …..x a)[m+n times]
= am+n (Example: 35 x 37 = 312)
Similar reasoning gives the following results:
am÷an = am-n (Example: 25÷22 = 23)
(am) n = amn (that is, am x n) (Example: (32)3 = 36)
If a ≠ 0 then a0 = 1 (Example: 100 =1 Note that100 equals, for example, 102÷102 which obviously equals 1.)
(ab)n = an bn (Example: (3 x 2)4 = 3 x 2 x 3 x 2 x 3 x 2 x 3 x 2 = 34 x 24)
a-n = 1÷an (= 1/an) (Example: 2-3 = 1/23 = 1/8 because, for example, 22÷25 = (2 x 2)÷(2 x 2 x 2 x 2 x 2) = 1÷(2 x 2 x 2) = 1/8)
But from above, 22÷25 = 22-5 = 2-3 So 2-3 = 1/23 = 1/8
