Lesson 1 MULTIPLYING MONOMIALS. What are we going to do… Multiply monomials. Simplify...

Lesson 1MULTIPLYING MONOMIALS

What are we going to do…

Multiply monomials.

Simplify expressions involving powers of monomials.

Monomial

A monomialmonomial is a number, a variable, or a product of a number and one or more variables.

An expression involving the division of variables is notnot a monomial.

Monomials that are real numbers are called constantsconstants.

Examples of Monomials

1. -5

2. x

3. abc3

4. 5xy2

7

Not a monomial: 4cd3

9abWhy?

Rule #1:Product of Powers

To multiply two powers that have the same base, add the exponents.

For any number x, xm(xn) = xm+n.

x12 ● x5 = x17

Example 1

(5x6)(x3)

= 5(x6x3)

= 5x9

Example 2

(4ab4)(-5a2b3)

= (4)(-5)(aa2)(b4b3)

= -20a3b7

Example 3…on your own!

(2ab5)(-a2b)

= (4)(-1)(aa2)(b5b)

= -4a3b6

Rule #2:Power of a Power

To find the power of a power, multiply the exponents.

For any number a, (am)n = amn.

(a4)3 = a12

Example 4

[(a2)3]2

= [a6]2

= a12

OR….

= [a] 2*3*2 =a12

Example 5

(x2)4

= x2*4

= a8

Example 6…on your own!

[(32)3]2

= [36]2

= 312

= 531,441

Rule #3:Power of a Product

To find the power of a product, find the power of each factor and multiply.

For all numbers x and y, (xy)m = xmym.

(-2x2y3)3 = (-2)3(x2)3(y3)3 = -8x6y9

Think of it like distributing the exponent!

Example 7

(3ab)3

= (3)3(a)3(b)3

= 27a3b3

Example 8

(5x2yz3)2

= (5)2(x2)2(y)2(z3)2

= 25x4y2z6

Example 9…on your own!

(-2x3yz4)2

= (-2)2(x3)2(y)2(z4)2

= 4x6y2z8

Simplifying Monomial Expressions

To simplify an expression involving monomials, write an equivalent expression in which:

1. each base appears exactly once

2. there are no powers of powers

3. all fractions are in simplest form

Example 10

[(8g3h4)2]2(2gh5)4

= [(8)2(g3)2(h4)2]2(2)4(g)4(h5)4

= [64g6h8]2(16g4h20)

= (64)2(g6)2(h8)2(16g4h20)

= 4096g12h16(16g4h20)

= (4096)(16)(g12g4)(h16h20)

= 65,536g16h36

Example 11

(ab4)(ab2)

= (aa)(b4b2)

= a2b6

Example 12…on your own!

(-4c4d4)(4cd)

= (-4)(4)(c4c)(d4d)

= -16c5d5

Example 13

(5a2b3c4)(4a2b4c3)

= (5)(4)(a2a2)(b3b4)(c4c3)

= 20a2b7c7

Example 14

(7pq7)2

= (7)2(p)2(q7)2

= 49p2q14

Example 15…on your own!

(5x3)2

= (5)2(x3)2

= 25x6

Example 16

(4cd)2 (-2d2)3

= (4)2(c)2(d)2 (-2)3(d2)3

= 16c2d2 (-8d6)

= (16)(-8)(c2)(d2d6)

= -128c2d8

Example 17

(2ag2)4 (3a2g3)2

= (2)4(a)4(g2)4 (3)2(a2)2(g3)2

= 16a4g8(9a4g6)

= (16)(9)(a4a4)(g8g6)

= 144a8g14

Review

1. When multiplying powers with the same base, we ______ the exponents.

2. When raising a power to a power, we _____________ the exponents.

add

multiply