Be able to multiply monomials. Be able to simplify expressions involving powers of monomials.
Monomials - a number, a variable, or a product of a number and one or more variables. Constant – A monomial that is a real number. Power – An expression in the form x n. Base – In an expression of the form x n, the base is x. Exponent – In an expression of the form x n, the exponent is n. Exponent Base Power
Definitions Product of Powers For any number a, and all integers m and n, a m * a n = a m+n. (a 3 ) (a 4 ) = a 3+4 = a 7 Power of a Power For any number a, and all integers m and n, (a m ) n = a mn. (a m ) n = a mn. Product of a Product For all numbers a and b, and any integer m, (ab) m = a m b m. (2*x) 2 = 2 2 x 2
Definitions Power of a Monomial For all numbers a and b, and all integers m, n, and p, (a m m n ) p = a mp b np. (2 2 x 3 ) 4 = 2 2*4 x 3*4 = 2 8 x 12 Quotient of Powers For all integers m and n, and any nonzero number a,
Definitions Zero Exponent For any nonzero number a, a 0 = 1. 4 0 = 1 Negative Exponents For any nonzero number a and any integer n,
Writing Using Exponents Rewrite the following expressions using exponents. The variables, x and y, represent the bases. The number of times each base occurs will be the value of the exponent. 2 3 2 3 2 3 2 3 3 4 2
Writing Out Expressions with Exponents Write out each expression the long way. The exponent tells how many times the base occurs. If the exponent is outside the parentheses, then the exponent belongs with each number and/or variable inside the parentheses.
Simplify the following expression: (5a 2 )(a 5 ). Step 1: Write out the expressions the long way or in expanded form. 5 5 25 aa aaaaaaa Step 2: Rewrite using exponents. 5 7 5 aaaaaaa a For any number a, and all integers m and n, a m a n = a m+n
Simplify the following: First, write the expression in expanded form. x 3 4 x 3 x 3 x 3 x 3 However, Therefore, Note: 3 x 4 = 12. For any number, a, and all integers m and n,
Simplify: (xy) 5 xy 5 xy 55 xxxxx ()( yyyyy) For all numbers a and b, and any integer m,
Simplify: 4 11 44 56 4 56 Apply the Product of Powers property ()a 36 a 18 a 36 Apply the Power of a Power Property. 3 4 3 xy 3 34 3 3 xy 27 123 xy Apply the Power of a Product Property and Simplify.