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Basic Definitions Exponent - tells how many times the base is used as a factor. Base – the number or variable that undergoes repeated multiplication. 2 BASE 4 EXPONENT means 2 2 2 2 16 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 Expressions x 5 means (use parentheses for multiplica tion ) ( x)( x)( x)( x)( x) y 3 means ( y )( y )( y ) Here are a few reminders when evaluating exponents • A base with an exponent of 0 equals 1 • 100 = 1 • 25,000,0000 = 1 • A base with an exponent of 1 equals the base number • 51 = 5 • 1,000,0001 = 1,000,000 • A positive base with a positive exponent equals a positive number. • 52 = 25 • (1/2)2 = ¼ • A negative base with an even exponent equals a positive number. • (-3)2 = (-3)(-3) = 9 • A negative base with an odd exponent equals a negative number. • (-3)3 = (-3)(-3)(-3) = -27 • A base with a negative sign in front equals a negative number. • -33 = -(3 *3*3) = -27 • -92 = -(9*9) = -81 Examples: • 2 2 ( ) 3 2 3 2 3 – Write out 𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 twice... ( ) * ( ) – Multiply your numerators – Multiply your denominators – Final Answer = • Example: 4 9 1 4 ( ) 2 – Write the fraction 4 times – Multiply the numerators – Multiply the denominators – Final Answer = 1 16 Multiplying Numbers with Exponents • To multiply numbers with exponents that have the same base, add the exponents and keep the base the same. • Example: 2ᵌ ∙ 2⁷ = (2 x 2 x 2) (2 x 2 x 2 x 2 x 2 x 2 x2) 2ᵌ +2 ⁷ = Answer = 29 = 512 28 Answer = 256 You Try: 4ᵌ ∙ 4⁴ Answer = 47 = 16, 384 Another example: 6⁵ ∙ 6 Answer = 66 = 66,656 MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base. 2 2 3 3 4 ( x )( x ) 5 (2 2 2)(2 2 2 2 2) ( x)( x)( x) ( x)( x)( x)( x) 28 x7 256 MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base. (52 )3 (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 ) x5 (43 )( x 2 )3 x5 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 ab MULTIPLY THE EXPONENTS POWER OF PRODUCT ( xy ) x y a a a Dividing Exponents 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 of Exponents Examples • 54 5 5*5*5*5 5 Simplify: the numerator has 53 and nothing left in the denominator. 53 = 125 • 3 3⁴ 3 3 *3*3*3 Simplify: the numerator has nothing left and the denominator has 33 . Answer = 1 27 Negative Exponents Negative Exponents Negative Exponents DIVISION PROPERTIES 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 3 x y (3x y ) 27 x y 3x y 2 8 x 3 y12 8 y6 9 6 27 x 6 27 x y