# Simplifying Expressions Containing only Monomials

We call monomials to those algebraic expressions consisting of a single term (although this term may contain subexpressions which do contain more than one term). In the following excercises we concentrate on the simplification of simple products, quotients, and powers – in fact, we are reviewing and illustrating the laws of exponents, but using algebraic expressions rather than simple numerical expressions.

We could say that "the rules" for combining powers are the following:

Usually simplification involves combining rules (iv) and (v) with one or more of rules (i), (ii), or (iii). When division is involved, rule (ii) brings in something like cancelling common factors between the numerator and the denominator of a fraction.

Remember that multiplication with simple numbers or symbols representing simple numbers is commutative – the order of the factors doesn’t matter:

a Â· b = b Â· a

Example 1:

Simplify

solution:

Property (iv) can be applied to products of more than two factors. All we need to remember to do is to apply the power outside the brackets to every factor inside the bracket. So

Thus, we get

Rule (i) can be used with a product of more than two factors as well.

Example 2:

Simplify

solution:

Separate all products into factors involving either a number or a power of a single symbol:

Now, rule (i) applies only when the powers being multiplied are powers of the same number or symbol – represented by the symbol ‘a’ in the statement of the rule at the beginning of this document. So, to position the factors in our expression for a potential simplification using rule (i), we rearrange individual factors to get matching symbols in sequence:

Note that in the above work, we used the fact that x = x 1 . Thus, for example

is a product of two factors of x. So, multiplying x 2 by x means the result has an additional factor of x, or three factors of x, hence needs to be written as x 3 . This is what results when we replace x by x 1 and then apply rule (i) for multiplying two powers together. Take care when the expression being simplified has a leading minus sign.

Example 3:

Simplify

solution:

The easiest way to take correct account of the leading minus sign is to keep it attached to the 5 in this case. Then, separating powers of each symbol and rearranging the order of the factors appropriately as we did in the previous example, we get

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