Laws of Exponents

(i) multiplication of two powers:

if c is any number, then

(ii) division of one power by another:

if c is any nonzero number, and m is a larger number than n, then

if c is any nonzero number, and m is a larger number than n, then

(iii) raising a power to a power:

if c is any number, then

(iv) raising a product to a power:

if c and d are any numbers,

(v) raising a quotient or a fraction to a power:

if c is any number, and d is any nonzero number, then

(vi) A Caution!

The five “laws” of exponents summarized in the square boxes above all involve multiplication or division only. As soon as addition or subtraction occurs, the obvious relations are NOT TRUE!

For example,

( 2 + 3 ) ⁴ = 5 ⁴ = 625

but

( 2 + 3 ) ⁴ 2 ⁴ + 3 ⁴,

since this latter expression evaluates to 16 + 81 = 97, which is clearly incorrect. Another example is

( 7 - 4 ) ⁵ = 3 ⁵ = 243

but

( 7 - 4 ) ⁵ 7 ⁵ - 4 ⁵,

since the latter gives 16807 – 1024 = 15783, which is clearly wrong. In both of these cases, the rules of priority for operations (described just ahead in these topic notes) are followed in the correct forms. It is necessary to evaluate the quantity in brackets first, and then apply the exponent to the result. The exponents here apply to the result of doing whatever operations are shown inside the brackets. This will become a very important rule to remember when the brackets contain symbolic expressions rather than simple numerical expressions.

So, for powers of sums and differences, there is no simple law (such as forms (iv) and (v) above for powers of products and quotients, respectively) that allows us to express them in terms of powers of the original terms in the sums or differences themselves:

All of the properties of powers described in this section will become even more useful when we deal with expressions involving symbols rather than just numbers.