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RATIONAL EXPONENTS Laws of Exponents Multiplication Properties •Product of powers •Power to a power •Power of a product Zero and negative exponents Division properties of exponents •Quotient of powers •Power of a quotient Basic Terminology Exponent In Exponential notation, the number of times the base is used as a factor. Base In Exponential notation, the number or Variable that undergoes repeated MULTIPLICATION . 2 4 EXPONENT means 2 2 2 2 16 BASE Important Examples IMPORTANT EXAMPLES 3 means (3 3 3 3) 81 4 (3) 4 means (3) (3) (3) (3) 81 3 means (3 3 3) 27 3 (3) means (3) (3) (3) 27 3 Variable Examples Variable Expressions x 5 means (use parentheses for multiplica tion) ( x)( x)( x)( x)( x) y 3 means ( y)( y)( y) Substitution and Evaluating STEPS 1. Write out the original problem. 2. Show the substitution with parentheses. 3. Work out the problem. Example : Solve if x 4; Solve if x 4; (4) 3 x3 x3 = 64 More Examples Evaluate the variable expression when x = 1, y = 2, and w = -3 ( x) ( y ) 2 2 ( x y) Step 1 ( x) ( y ) 2 2 ( x y) Step 1 2 wx y Step 2 Step 2 (1) (2) 2 Step 3 1 4 5 wx y Step 1 Step 2 (1) 2 (2) 2 2 Step 3 (3) 9 2 (3)(1) 2 Step 3 (3)(1) 3 MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base. 2 2 3 5 3 4 ( x )( x ) (2 2 2)(2 2 2 2 2) ( x)( x)( x) ( x)( x)( x)( x) 28 256 x7 MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base. 2 3 (5 ) (x 2 ) 4 (52 )(52 )(52 ) ( x 2 )( x 2 )( x 2 )( x 2 ) 56 x8 MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions. (6 5) 2 (5 xy) 3 (6) 2 (52 ) 3 3 3 (5 )( x )( y ) (4 x 2 )3 x 5 (64)( x 6 ) x 5 (43 )( x 2 )3 x 5 64x11 36 25 900 125 x 3 y 3 (64) ( x 2 )( x 2 )( x 2 ) x5 MULTIPLICATION PROPERTIES SUMMARY PRODUCT OF POWERS x x x a b a b ADD THE EXPONENTS POWER TO A POWER x a b x a b MULTIPLY THE EXPONENTS POWER OF PRODUCT DISTRIBUTE THE EXPONENT ( xy) x y a a a ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1. 33 27 32 9 31 3 30 1 1 1 3 1 3 3 1 1 2 3 2 3 9 1 1 33 3 3 27 1 1 2 2 x 2 2 2 x x 1 1 1 2 (2 x) 2 2 2 2 (2 x) 2 x 4x 2 DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base. x 5 ( x)( x)( x)( x)( x) ( x)( x) 2 x 3 x ( x)( x)( x) 1 DIVISION PROPERTIES POWER OF A QUOTIENT 4 x ( x 2 )4 x8 3 3 4 12 (y ) y y 2 Hard Example 3 3 3 6 2 3 3 12 2 xy 2 x y ( 2 xy ) 8 x y 3 4 3 9 12 3 4 3 9 6 (3x y ) 3 x y 27 x y 3x y 2 8 x 3 y12 8 y6 9 6 27 x 6 27 x y Summary of Zero, Negative, and Division Properties ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power ( x) 1 Negative Exponents x a 1 a x and 1 a x a x 0 Quotient of powers xa a b x b x Power of a quotient a x x a y y a