12.01 Multiplying Monomials

A monomial is a number, a variable, or a product of both. Examples: 8, x, 5y, x 3, 4x 2, – 6xy 7 Exponential Notation amam a is called the base. m is called the exponent or power. The exponent indicates how many times to multiply the base by itself

x 3 means amam a · a · a · a ···· a m times |_______________| = x · x · x 9x 2 y 2 = 9 · x · x · y · y Write w · w · w · w · w using exponents.= w 5 Write 2 · y · y · y · 3 using exponents.= 6y 3 Write x · x · y · y · x using exponents.= x 3 y 2 (5w) 2 8y 4 = 8 · y · y · y · y = 5w · 5w Write 4x · 4x · 4x using exponents.= (4x) 3 or 64x 3

Simplify x 3 · x 2 = x · x · x= x 5 · x Simplify 2w 4 · 8w 2 = 2 · w · w · w · w· 8 · w · w = 16w 6 We want to find a rule that eliminates all these steps. When multiplying monomials, add the exponents of the variables that have the same base. a m · a n = a m + n If there are coefficients, multiply the coefficients first.

Simplify the following expressions. x 5 · x 3 =x a m · a n = a m + n = x 8 y 7 · y – 3 =y 7 + (– 3) = y 4 x 4 y 6 · xy 8 =x y = x 5 y 14 Important x = x 1 3x 5 · 5x 9 =(3 · 5)(x 5 · x 9 ) = 15x 14 – 6x 5 y · 9xy 3 =(– 6 · 9)(x 5 · x)(y · y 3 ) = – 54x 6 y 4

Simplify the following expressions x 9 · x = a m · a n = a m + n x 10 y 8 · y – 2 =y 6 x 3 y 5 · x – 3 y 8 =x 0 y 13 Important x = x 1 2x 4 · 8x 7 =16x 11 – 8xy · 7xy 2 =– 56x 2 y 3 Try This: x 6 y 2 · 3xy 4 =3x 7 y 6