To solve polynomial expressions, you may need to simplify monomials -- polynomials with only one term. Simplifying monomials follows a sequence of operations involving rules for handling exponents, multiplying and dividing. Always handle variables with exponents raised to a power first.
Definitions of Terms
The base is a variable, and an exponent is the power a variable is raised to. A variable with no visible exponent is assumed to have an exponent of 1. A variable with an exponent of zero is equal to the value 1. A coefficient is a number that precedes a variable and is a multiplier of that variable; for example, in 7y, the 7 is the coefficient.
Rules for Simplifying Monomials
The power of a power rule says that when evaluating a power of a power, multiply the exponents of base variables. The multiply monomials rule says that when you multiple monomial expressions, add the exponents of like bases. The dividing monomials rule says that when you divide monomials, subtract the exponents of like bases.
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The expression x^y means x to the y power, for example: 2^3 equals 2 times 2 times 2, which yields 8.
An example of simplifying monomials using the power of a power rule might be: [3x^3 y^2]^2 = 9x^6 y^4. If x = 2 and y = 3, on the left side of the equation, you have: 2^3 = 8, 3 times 8 = 24, 3^2 = 9, 9 times 24 = 216 and 216^2 = 46,656. On the right side of the equation, you have: x^6 = 64, 9 times 64 = 576, 3^4 = 81 and 81 times 576 = 46,656.