Powers and indices

Powers and indices

As you have already seen, when a number is the product of the same factor multiplied by itself this number is called a power of the factor. For example, we know that 3 X 3 = 9. Therefore, we can say that 9 is a power of 3. To be precise, it is the second power of 3, because two 3s are multiplied together to produce 9. Similarly, 16 is the second power of 4. We may use literal terminology to generalize the relationship between powers and factors.

So, the second power of a means a X a (or a ^. a), this is written as a², where a is known as the base (factor) and 2 is the index (or exponent). Thus writing the number 9 in index form we get 9 = 3², where 9 is the second power of 3, 3 is the base (factor) and 2 is the index (exponent).

The above ideal can be extended to write arithmetic numbers in index or exponent form, for example, 5² = 25, 9² = 81 and 3³ = 27. Notice that the second power of 5 gives the number 25 or 5 X 5 = 25 similarly, 3³ means the third power of 3, literally 3 X 3 X 3 = 27. The idea of powers and indices (exponents) can be extended to literal numbers. For example:

a ^. a ^. a ^. a ^. a or a⁵ or in general a^m, where a is the base (factor) and the index (or exponent) m is any positive integer. a^m means a used as a factor m times and is read as the ‘mth power of a’.

Note that since any number used as a factor once would simply be the number itself, the index (exponent) is not usually written, in other words `a' means a^l.

Now, providing the base of two or more numbers expressed in index (exponent) form are the same, we can perform multiplication and division on these numbers, by adding or subtracting the indices accordingly.

We will from now only refer to the index of a number rather than its exponent of a number as its index in order to avoid confusion with particular functions, such as the exponential function, which we will study later.