Multiplication and Division of Monomials

Overview:

Monomials can be multiplied and divided like any other algebraic expression. They can be simplified by applying the rules of exponents, if the bases are the same. It is not necessary to solve the equation for the values of the variables before operations can be performed.

Multiplying Monomials with More than One Variable:

Applying the definition of multiplication, a problem such as 3a(2c) can be simplified by multiplying 3a times 2c.  It is not necessary to solve for the values of a or c. Because of the Commutative Property of Multiplication, the factors can be rearranged, so that the problem becomes 3∙2∙a∙c, or 6ac. The result is still a monomial.

Multiplying Exponents:

If monomials are to be multiplied and the exponents have the same base, the exponents can be added to form the new monomial. Suppose the problem were 4a³(9a²). As before, use the Commutative Property of Multiplication to rearrange the factors, so that the problem becomes 4∙9∙a∙a∙a∙a∙a, or 36a⁵. If the problem were 2a²∙ 13b⁴, the exponents couldn’t be combined. The simplest form of the monomial would be 26a²b⁴.

Powers of Monomials:

If a monomial is raised to a power, it can be simplified with a single exponent. For example, (2x³)² actually means 2x³∙2x³, or 4x⁶. Similarly, (3y²z)⁴ can be simplified as (3⁴)(y²)⁴(z⁴) or 81y⁸z⁴. The general rule in algebraic terms is that (x^a)^b means x^ab. Also, (xy)^a means the same thing as x^ay^a, even when the monomial itself has a number of factors.

Dividing Monomials:

The definition of division is multiplying by the inverse of the denominator, so that (6x)/2 means the same as 6x ∙1/2 or 3x. If there are exponents in the numerator and the denominator, the exponents must be the same in order to simplify the expression. An expression such as 4x³/2x² can be simplified further as (4x³)(1/2x^-2) or (4∙1/2)(x³x^-2). One half of 4 is 2, and by the definition of multiplying exponents with the same base, x³x^-2 is the same thing as x^3-2 or x¹, or 2x. Another way to state the rule in algebraic terms is that if x is real and not equal to 0, then x^a/x^b means the same thing as x^a-b.

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