Monomials Multiplying Monomials and Raising Monomials to Powers

Vocabulary Monomials - a number, a variable, or a product of a number and one or more variables Constant – a monomial that is a number without a variable. 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. 4x, 20x 2 yw 3, -3, a 2 b 3, and 3yz are all monomials

Writing Expressions without Exponents Write out each expression without exponents (as multiplication): 40 a 3 b 6 = 40aa b b b b b b (x y ) 5 = x y a.... or

Writing Expressions without Exponents Write out each expression without exponents (as multiplication): 40 a 3 b 6 = 40aa b b b b b b (x y ) 5 = x y a.... or = x x x x x y y y y y = x5x5 y5y5

Simplify the following expression : (5a 2 )(a 5 ) Product of Powers (5a 2 )(a 5 ) = Step 1: Write out the expressions in expanded form 5 a a a a a a a Step 2: Rewrite using exponents. (5a 2 )(a 5 ) =5 a. = 5a7a7 7

For any number a, and all integers m and n, a m a n = a m+n. Product of Powers Rule 1. x 5 x 7 =. x (m)(m 5 )(m 2 )= m 8 3. (x 3 )( y 2 ) = x 3 y 2

Multiplying Monomials Remember when multiplying monomials, you ADD the exponents. 1) x 2 x 4 x 2+4 x 6 2) 2a 2 y 3 3a 3 y 4 2 3a 2 a 3 y 3 y a 2+3 y a 5 y 7

Simplify m 6 (m)(m) 1.m 7 2.m 8 3.m 12 4.m 13

If the monomials have coefficients, multiply those, but still add the powers. Multiplying Monomials 1. (10 x 2 ) (-3x 5 ) = -30x 7 2. (4a -2 ) (2a 6 ) = 8a m 1/2 2m 1/2 = 4m 1 = 4m

These monomials have a mixture of different variables. Only add powers of like variables. Multiplying Monomials 1. (10a 3 b) (5a 8 b 4 ) =50 11 a b 5 2. (-rt) (3r 2 ) =-3r 3 t 3. (2d 5 e 4 f) (3d 2 ef 3 ) (3d -4 e 4 f 2 ) = 18 d 3 e 9 f 6

Simplify the following: ( x 3 ) 4 Power of Powers Step 1: Write out the expression in expanded form. Step 2: Simplify, writing as a power. (x 2 ) 3 = x2x2 x2x2 x2x2 = x x x x x x (x 2 ) 3 = x 6 Note: 2 x 3 = 6

Power of Powers Rule For any number, a, and all integers m and n, In other words, when you have a power over a power, multiply to exponents a. (x 4 ) 5 = x b. (y 3 ) 10 = y c. (m 4 n 2 ) 3 = m n

Power of a Power When you have an exponent with an exponent, you multiply those exponents. 1) (x 2 ) 3 x 2 3 x6x6 2) (y 3 ) 4 y 12

Simplify (p 2 ) 4 1.p 2 2.p 4 3.p 8 4.p 16

Power of a Product When you have a power outside of the parentheses, everything in the parentheses is raised to that power. 1) (2a) 3 23a323a3 8a 3 2) (3x) 2 9x 2

Simplify (4r) r r r r 4

Power of a Monomial This is a combination of all of the other rules. 1) (x 3 y 2 ) 4 x 3 4 y 2 4 x 12 y 8 2) (4x 4 y 3 ) x 4 3 y x 12 y 9

Simplify (3a 2 b 3 ) a 8 b a 6 b a 16 b a 8 b 12