EXPONENTS AND THEIR PROPERTIES

Transcription

1 EXPONENTS AND THEIR PROPERTIES Bsic Definitions nd Nottion mens or 8 The number is the eponent. The number is the bse. mens is the eponent; is the bse. WARNING! The bse of n eponent is the symbol directly in front of the eponent. - mens - ( ) The symbol directly in front of the eponent is. (-) mens (-)(-)(-) The symbol directly in front of the eponent is prenthesis. The eponent belongs to everything in the prentheses. mens ( ) () mens ()()()() - mens - ( ) 1 mens 1 A negtive eponent mens the reciprocl of the bse with positive eponent. It does not indicte negtive number! The negtive sign in front remins. The negtive eponent produces reciprocl. Properties of Eponents 1. b + b Product Rule: When multiplying quntities with the sme bse, keep the common bse nd dd the eponents. Emple: ( ) ( ) ( ) 7 +. b Quotient Rule: When dividing quntities with the sme bse, keep the common ( 0) bse nd subtrct the denomintor eponent from the numertor eponent. Emple: 6 / / 6- / / Deprtment of Mthemtics, Sinclir Community College, Dyton, OH 1

2 . 0 1 Zero Eponent: Any bse (ecept 0) rised to the zero power equls 1. ( 0) Emples: ( y z) 0 1. Power Rules ) ( ) b b If n eponentil epression is rised to nother power, keep the bse, multiply the eponents. Emple: ( ) ( ) ( ) ( ) ( )( )( ) 6 ( ) Wrning: ( ) nd re different situtions with different eponent rules! Lern to tell them prt! ( ) 6 + b) (y) y If you hve product being rised to power, you cn rise ech fctor in the product to tht power. Emple: () ()()() ()()()()()() or 8 c) y y If you hve quotient being rised to power, you cn rise both the numertor nd the denomintor to tht power. Emple: Negtive Eponents: 1 ) A negtive eponent mens the reciprocl of the bse with positive eponent. Emples: (yz) 1 (yz) 1 1 (y) (y) y 7y Deprtment of Mthemtics, Sinclir Community College, Dyton, OH

3 - b b) y The fctor in the numertor with negtive eponent moves to the -b y denomintor. The fctor in the denomintor with negtive eponent moves to the numertor. Both eponents become positive. Emples: y y w w z w z w z y z y Wrning: This only works with fctors. It will not work if sums or differences re involved. + y z 1 cnnot be inverted to the form z 1 + y! - c) y A quotient with negtive eponent outside the prentheses my be y inverted inside the prentheses, with positive eponent on the outside. Emples: y z z y More Emples nd Using the Properties Together Simplify the following: 1) y The properties of eponents pply only when the bses re the sme. Strt by grouping like bses together. y Use the product rule nd dd eponents. + y 7 y Cnnot be simplified further (bses re not the sme). Deprtment of Mthemtics, Sinclir Community College, Dyton, OH

4 ) ( ) Use the power rule to distribute the eponent nd squre ech fctor of the product. ( ) Use the power rule (rising to second power) nd multiply eponents. 6 Note: If your bse is number, you my multiply it out if you wish or you my 6 leve it in eponentil form. Either form is cceptble. As rule of thumb, multiply it out if it s esy, leve it in eponentil form if it s too much work. ) - ( ) Tricky - be creful! Strt with the ( ) nd simplify. - 1 Both bses re s (the negtive sign is not involved) so you cn simplify further. - ( 1 ) - ( 1 ) - 1 ) 0 7 y z 17 yz This looks like difficult problem. Resist the urge to simplify inside the prentheses first. Look t the eponent outside those prentheses. 1 Any bse (ecept 0) rised to the zero power equls 1. ) You my rise ech fctor in numertor nd denomintor to the fifth power or you my simplify inside the prentheses first. ( ) ( ) 0 1 OR i i 0 1 i Cnnot be simplified further; bses re not the sme. Deprtment of Mthemtics, Sinclir Community College, Dyton, OH

5 6) Since the bses re the sme, we don t need to use ny negtive eponent properties for this problem. Just use the quotient rule. (- ) + 9 7) ( y) Use the negtive eponent rules to invert the quotient with positive 7 eponents. y ( ) ( y ) 7 or ( ) ( ) ( ) y 7 1 ( y) ( y) Remember: ( + b) m m + b m! ( y) cnnot be further simplified. 8) 1 1 y Here's jumble of positive nd negtive eponents ll mied up. A simple wy to proceed is to first rise ll the inner fctors, y including those with eponents, to their outer eponents. 1 ( ) ( ) ( )( ) ( y ) ( )( ) ( y ) 1 1 Use the power rule to multiply eponents. 1 6 y Group numbers nd like bses in the numertor. Multiply numbers 6 y nd use the product rule on the vribles to simplify. * y Seprte terms with positive eponents from those with negtive 6 y eponents y 1 1 i i Use the negtive eponent property to crete reciprocls with 6 y positive eponents y y i i Use the product nd quotient rules on like vribles i9 y 60 y (6 ) (6+ 6) 1 * The quotient rule could hve been pplied t this step on the vribles. Deprtment of Mthemtics, Sinclir Community College, Dyton, OH

6 Problems Simplify: 1) 9 ) (7y) (7y) y ) ( ) ) y ) 8 10 y z 10 y z 6) (y ) 7) z z 8) y 9) y z y 10) 10 y z 8z ) + 1) (m) (m) - 1) 1 b c 1) (7 6 z ) - Answers 1) 1 ) 7 7 y 7 or (7y) 7 ) 1 ) y ) z 6) y 1 or 8 y 1 7) z or z 8) 8 y 1 16 or 8 y 1 y z 9) 10 or 16 y z ) 1 11) + (Trick question. nd re not like terms nd cnnot be dded. There re no eponent properties tht pply to this sitution!) 1) m 1) c b 1) 18 z 1 Deprtment of Mthemtics, Sinclir Community College, Dyton, OH 6