2. © William James Calhoun, 2001 9-1: Multiplying Monomials To start the chapter, a couple of terms need to be defined. monomial - a number, a variable, or a product of a number and one or more variables Examples: constant - either a monomial that is only a real number (no variables attached) -or- can refer to the real number part of the monomial (also called coefficient later) OBJECTIVES: You will multiply monomials and simplify expressions involving powers of monomials. MonomialsNot Monomials 4x 3 q 11aba + b 5 - 7d 12

3. © William James Calhoun, 2001 9-1: Multiplying Monomials Remember what “2 5 ” really means and why its used? In dealing with math, there were instances where a common base (the 2) was multiplied by itself several times, like: 2 x 2 x 2 x 2 x 2. Rather than write that out every time, mathematicians created the exponential notation where 2 5 means two multiplied by itself five times. It is now time to put the exponential notation on variables. In this section we will learn four rules about dealing with exponents.

4. © William James Calhoun, 2001 9-1: Multiplying Monomials If “2x2x2x2x2” can be written as 2 5, what do you think x 6 represents? From the defining of exponents, we can see that x 6 represents some unknown number multiplied by itself six times. In long-winded terms: x 6 = (x)(x)(x)(x)(x)(x) This part is easy and is the same as dealing with plain old numbers. Now, what happens when letters and powers start combining?

5. © William James Calhoun, 2001 9-1: Multiplying Monomials Using the definition of what an exponent is, examine: x 5 y 4 (4x 3 ) In long terms, this problems becomes: (x)(x)(x)(x)(x)(y)(y)(y)(y)(4)(x)(x)(x). From this, we can get all the x’s lined up with each other since order does not matter when multiplying (commutative property) : (x)(x)(x)(x)(x)(x)(x)(x)(y)(y)(y)(y)(4). Now, move the constant to the front and redo the power notation realizing there are eight x’s multiplied together and four y’s multiplied together: 4x 8 y 4. Do you need to go through all those steps every time you hit one of these problems? Absolutely... There is a simple rule you need to memorize which handles this.

6. © William James Calhoun, 2001 9-1: Multiplying Monomials Another way to put this rule is: “When same bases are multiplied, add their exponents.” So, x 5  x 6 = x (5 + 6) = x 11. To handle the first example, remember you can change order within the monomial. You will want to get all constants and same bases together. Multiply constants and add exponents of same bases. NEVER add powers of different bases! For any number a, and all integers m and n, a m  a n = a (m + n). 9.1.1 PRODUCT OF POWERS

7. © William James Calhoun, 2001 9-1: Multiplying Monomials EXAMPLE 1: Simplify each expression. A. (3a 6 )(a 8 )B. Not that you should do this first step all the time, but in a technical sense, this problem reads: 3(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a) So, how many a’s are multiplied by each other? 14 So, to write this simply, we say… 3a 14 In big problems like this, everything is multiplied and can be written in any order. Put all the constants and like variables together. 8(-3)(y 3 )(y 2 )(y 4 )(x 2 )(x) { Use calculator to multiply constants. { Count how many x’s are multiplied together. { Count how many y’s are multiplied together. -9x3x3 y9y9 -9x 3 y 9 Just remember the rule: Same base multiplied, add exponents, so: a 6 (a 8 ) = a (6 + 8) = a 14. This makes the problem easy.

8. © William James Calhoun, 2001 9-1: Multiplying Monomials When you have a power to a power, multiply the powers. So, (x 4 ) 3 = x 4(3) = x 12. This is like distribution. Everything inside gets a piece of what is on the outside. So, (xy) 3 = x 3 y 3. For any number a, and all integers m and n, (a m ) n = a mn. 9.1.2 POWER OF A POWER For any numbers a and b, and any integer m, (ab) m = a m b m. 9.1.3 POWER OF A PRODUCT

9. © William James Calhoun, 2001 9-1: Multiplying Monomials When you are asked to simplify an expression, you must rewrite it so: (1) there are no powers of powers left, (2) each base appears only once (no repeats on letters), and (3) all fractions are reduced. This is pretty much the same as the last rule. So, (x 2 y) 3 = x 2(3) y 1(3) = x 6 y 3. For any numbers a and b, and any integers m, n, and p, (a m b n ) p = a mp b np. 9.1.4 POWER OF A MONOMIAL

10. © William James Calhoun, 2001 9-1: Multiplying Monomials ( ) ( )( ) EXAMPLE 2: Simplify (2a 4 b) 3 [(-2b) 3 ] 2. First thing is use the Power of Products rule. Use the Power of a Power rule. 3x2 Take care of the constants and use Power of a Power rule. 8 4x3 a 12 b3b3 Use the Power of Products rule. (-2) 6 b6b6 Get constants and variables together. Also handle the (-2) 6. (8)(64)(a 12 )(b 3 )(b 6 ) And a partridge in a pear tree. Twelve a’s. Nine b’s.8x64 (512)(a 12 )(b 9 ) 512a 12 b 9

11. © William James Calhoun, 2001 9-1: Multiplying Monomials HOMEWORK Page 499 #17 - 35 odd