Monomials Multiplying Monomials and Raising Monomials to Powers

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 4 x, 20 x 2 yw 3 , -3, a 2 b 3 , and 3 yz are all monomials • Constant – a monomial that is a number without a variable. • Base – In an expression of the form xn, the base is x. • Exponent – In an expression of the form xn, the exponent is n.

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

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

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

Product of Powers Rule For any number a, and all integers m and n, am • an = am+n. 1. x 2. 5 . 7 x = x 12 8 m 5 2 (m)(m )= 3. (x )( 3 y 2 3 ) =x y 2

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

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

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

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

Power of Powers Simplify the following: ( x 3 ) 4 Step 1 : Write out the expression in expanded form. 2 3 (x ) Step 2 : 2 • x 2 x = = x • x • x • x Simplify, writing as a power. 2 3 (x ) =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. b. c. 4 5 (x ) = 3 10 (y ) x = 4 2 3 (m n ) 20 y = 30 12 6 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 x 6 2) (y 3)4 y 12

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

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

Simplify (4 r)3 1. 2. 12 r 4 3. 64 r 3 4. 64 r 4 3 12 r

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) (4 x 4 y 3)3 43 x 4 • 3 y 3 • 3 64 x 12 y 9

Simplify (3 a 2 b 3)4 1. 2. 3. 4. 12 a 8 b 12 6 7 81 a b 81 a 16 b 81 81 a 8 b 12