Section P.2: Rules of Exponents If a > 0, b > 0, and m,n are integers, the following are true for all real numbers x and y: 1. 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 Example: 𝑥 4 ∙ 𝑥 3 = 𝑥 4+3 = 𝑥 7 2. 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 Example: 𝑦 5 𝑦 2 = 𝑦 5−2 = 𝑦 3 3. ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛 Example: 𝑥 2 3 = 𝑥 6 4. 𝑎𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 Example: 𝑥𝑦 2 = 𝑥 2 𝑦 2 5. 𝑎 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 Example: 3 7 3 = 3 3 7 3 = 27 343

More Rules of Exponents 𝑎 −𝑚 = 1 𝑎 𝑚 Example: 2 −3 = 1 2 3 = 1 8 1 𝑎 −𝑚 = 𝑎 𝑚 Example: 1 𝑥 −4 = 𝑥 4 𝑎 −𝑚 𝑏 −𝑛 = 𝑏 𝑛 𝑎 𝑚 Example: 3 −2 𝑥 −5 = 𝑥 5 3 2 = x 5 9 𝑎 0 =1 Example: 2𝑥𝑦 0 =1

Simplify the Exponential Expressions 1. −3𝑎 𝑏 4 4𝑎 𝑏 −3 =−3∙4∙𝑎∙𝑎 ∙ 𝑏 4 ∙ 𝑏 −3 =−12 𝑎 2 𝑏 2. 2𝑥 𝑦 2 3 = 2 3 𝑥 3 𝑦 2 3 =8 𝑥 3 𝑦 6 3. 5 𝑥 3 𝑦 2 = 5 𝑥 3 2 𝑦 2 = 5 2 𝑥 3 2 𝑦 2 = 25 𝑥 6 𝑦 2

Simplify the Exponential Expressions 1. 1 3 𝑥 −2 = 1 𝑥 2 3 = x 2 3 2. 12 𝑎 3 𝑏 −4 4 𝑎 −2 𝑏 = 3 𝑎 3 𝑎 2 𝑏 𝑏 4 = 3 𝑎 5 𝑏 5 3. 3 𝑥 2 𝑦 −2 = 3 𝑥 2 −2 𝑦 −2 = 3 −2 𝑥 2 −2 𝑦 −2 = 3 −2 𝑥 −4 𝑦 −2 = 𝑦 2 3 2 𝑥 4 = 𝑦 2 9 𝑥 4

Dividing out vs. Subtracting Exponents Simplify: 𝑥 5 𝑥 2 = 𝑥 5−2 = 𝑥 3 Different approach: 𝑥 5 𝑥 2 = 𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 = 𝑥 3 Simplify: 𝑦 3 𝑦 7 = 𝑦𝑦𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦 = 1 𝑦 4 Simplify: 𝑥 7 𝑦 5 𝑥 3 𝑦 9 = 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑦𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 = 𝑥 4 𝑦 4

Simplify the Exponential Expressions 1. 𝑥 2 𝑥 𝑛 𝑥 3 𝑥 𝑛 = 1 𝑥 2. 𝑧+2 −3 𝑧+2 −1 = 𝑧+2 −4 = 1 𝑧+2 4 3. −2 𝑥 2 3 4 𝑥 3 −1 = −2 𝑥 2 3 4 𝑥 3 1 = −2 3 𝑥 2 3 4 1 𝑥 3 1 = −8 𝑥 6 4 𝑥 3 =−2 𝑥 3 4. 𝑎 −2 𝑏 −2 𝑏 𝑎 3 = 𝑎 −2 𝑏 −2 𝑏 3 𝑎 3 = 𝑎 −2 𝑏 3 𝑏 −2 𝑎 3 = 𝑏 2 𝑏 3 𝑎 2 𝑎 3 = 𝑏 5 𝑎 5

− 𝑥 2 vs. −𝑥 2 If a negative number in parentheses is squared, or raised to any even power, the answer is positive. If a number with a negative sign in front is squared, and there are no parentheses, the answer is negative. Examples: −4 2 =16 while − 4 2 =−16 −3 4 =81 while − 3 4 =−81 However… − 𝑥 3 = −𝑥 3 This is true for any odd power. Evaluate at x = -5: −6 𝑥 2 −4 𝑥 3 −6 −5 2 −4 −5 3 = −6 25 −4 −125 = −150+500=350

Fraction Exponents and Radicals 𝑎 1/𝑛 = 𝑛 𝑎 Example: 2 1/3 = 3 2 𝑎 𝑚/𝑛 = 𝑛 𝑎 𝑚 Example: 8 2/3 = 3 8 2 Rules of Radicals: 𝑛 𝑎 𝑛 𝑏 = 𝑛 𝑎𝑏 Example: 2 3 = 6 𝑛 𝑎 𝑛 𝑏 = 𝑛 𝑎 𝑏 Example: 3 4 3 9 = 3 4 9 𝑛 𝑎 𝑛 =𝑎 and 𝑛 𝑎 𝑛 =𝑎 Examples: 3 𝑥 3 =𝑥 and 5 2𝑥𝑦 5 =2𝑥𝑦

Simplify the radical expressions 5 20 = 5∙20 = 100 =10 4 4𝑥 𝑦 3 4 =4𝑥 𝑦 3 3 625 729 = 3 625 3 729 = 3 125∙5 9 = 3 125 3 5 9 = 5 3 5 9 10 243 −3 27 =10 81∙3 −3 9∙3 =10 81 3 −3 9 3 =10∙9 3 −3∙3 3 =90 3 −9 3 =81 3 5. 3 256 +3 3 108 = 3 64∙4 +3 3 27∙4 = 4 3 4 +3∙3 3 4 =4 3 4 +9 3 4 =13 3 4