Properties of Exponents Section 5-1

Objectives I can use all exponent rules to simplify expressions.

Properties of Exponents Negative Exponents Multiplying Exponents Dividing Exponents Properties of Powers Zero Exponent

Negative Exponents For any real number a; and any integer n; where a  0; then the following is true:

Negative Exponents You cannot have negative exponents in any of your final simplified answers. You will use the previous rule to convert all negative exponents into positive exponents: Ex: x-3 =

Multiplying Powers For any real number a; and integers m & n; am • an = am+n Example: x7 • x4 = x11

Dividing Powers For any real number a; and integers m & n; am  an = am-n Example: x7  x4 = x3

Properties of Powers If m & n are integers, then the following hold true: Power of a Power: (am)n = amn Power of Product: (ab)m = ambm Example 1: (x3)4 = x12 Example 2: (xy2)3 = x3y6

Zero Exponent Given any number m, where m  0 Then m0 = 1 Examples : (-89)0 = 1

Using Exponent Properties to Simplify Monomials Simplify (2x2y3)(-5x4y2) We can write this in simplest expanded form: (2•x • x • y • y • y)(-5 • x • x • x • x • y • y ) Using commutative property to regroup (2 •–5)(x • x • x • x • x • x)(y • y • y • y • y) -10 x6 y5

Examples 2-3 Simplify (a4)5 a20 Simplify (-5p2s4)3 (-5)3p6s12

GUIDED PRACTICE for Examples 1 and 2 2 3 3. 9 SOLUTION 2 3 23 = 9 93 8 2 3 9 SOLUTION 2 3 9 23 93 = Power of a quotient property 8 729 = Simplify and evaluate power.

GUIDED PRACTICE for Examples 3, 4, and 5 s 3 2 7. t–4 SOLUTION s 3 2 = Power of a product property t–8 s6 = Evaluate power. s6t8 = Negative exponent property

GUIDED PRACTICE for Examples 3, 4, and 5 8. x4y–2 3 x3y6 SOLUTION = (x4)3 (y–2)3 (x3)3(y6)3 Power of a powers property = x12y–6 x9y18 Power of a powers property = x3y–24 Power of a Quotient property x3 y24 = Negative exponent property

Simplify the expression. Tell which properties of exponents you used. GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. Tell which properties of exponents you used. 5. x–6x5 x3 SOLUTION x–6x5x3 = x–6x5 + 3 Power of a product property = x2 Simplify exponents.

GUIDED PRACTICE for Examples 3, 4, and 5 6. (7y2z5)(y–4z–1) SOLUTION Power of a product property = (7y2 – 4)(z5 +(–1)) Simplify = (7y–2)(z4) Negative exponent property = 7z4 y2

Homework Worksheet 8-1