Exponents. Learning Objectives 4-1
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1 Eponents -1 to - Learning Objectives -1 The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient We can simplify by combining the like terms. Eample: Simplify each epression, if possible. + + y y y y
2 Eponents denote repeated multiplication Eample: Find base eponent Eample: Identify the base of each epression and evaluate: (-8) -7 Eamples: Write each epression using eponents. π r r ( )( )( i)( i) Natural Number Eponents If n is a natural number, then n factors of n =. Eamples: 1 = ( + 1) = (y + 1)(y + 1) Eamples of factoring (-7a) = (-7a)(-7a)(-7a) The Product Rule for Eponents Multiplying eponents with the same base: = therefore, = =? 7 The product rule for eponents states: to multiply two eponential epressions with the same base, keep the common base and add the eponents. m n = m+n
3 Eamples Evaluate the following: 7 8 (7 7 ) (-6) (1/) 1 (-a b )(a b ) -8a 8 b 6 The Quotient Rule for Eponents State that when you divide like bases you subtract their eponents. If m and n represent natural numbers, m>n, and 0, then m n = m n NOTE: you always take the numerator s eponent minus your denominator s eponent, NOT the other way around. Eamples y y 11 y 8 7 a b a b -a b
4 The Power Rule for Eponents States when you raise a base to two eponents, you multiply those eponents together. If m and n represent natural numbers, then m mn ( ) n = Eample: ( z ) z 8 Powers of a Product and a Quotient States when you have a PRODUCT raised to an eponent, you can simplify by raising each base in the product to that eponent. If n represents a natural number, then (y) n = n y n n and if y 0, then = n y y n Eamples Simplify: (z) 16z 7 ( c d ) (-ab ) -7a b 1 c d c10 d 1
5 Laws of Eponents: For any integers m, n (assuming no divisions by 0) Laws of Eponents m n = m = n m = ( ) n ( ) n y = eamples n = y Eamples: Simplify each of the following. 9 6 ( yy) y h h 0 ( b ) Eamples: Simplify each of the following. ( uv )( uv 7 ) bb bb ( ) 6 ( t tt ).1 Natural Number Eponents
6 Eamples: Simplify each of the following. ( yy) ( y) ( y) 9 y.1 Natural Number Eponents Eamples: Simplify each of the following. 8 u v t ( a) ( a) 1 10 Eamples: Simplify each of the following. ( aa ) aa ( y ) ( y)
7 Eamples: Simplify each of the following. ttt tt Eamples: Evaluate using the rules for eponents. 10 ( 10)( 10 ) ( 10 ) Eamples: Find the area. Find the volume. 7 meters meters 8 cm cm cm Section.1 Review The product rule for eponents The quotient rule for eponents The power rule for eponents Power rules for products and quotient
8 Section - Learning Objectives Zero eponents Negative integer eponents Variable eponents Zero Eponents If represents any nonzero real number, then 0 = 1 0 = = = = 1 therefore, 0 = 1 Negative Integer Eponents If represents any nonzero number and n represents a natural number, then n 1 = n A negative eponent just means that the base is on the wrong side of the fraction line, so you need to flip the base to the other side.
9 Eamples (-) or Variable Eponents Eamples: 7 7 m m 1 t t 7m m a a m m 1 m a Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero ( ) 0 0
10 Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero. 0 yz y 0 11 ( 8) 1. Zero and Negative Integer Eponents Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero. 1 7 ( ) Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero. ( ) ( m n) 16 1
11 Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero t 10 t ( b ) ( b ). Zero and Negative Integer Eponents Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero b b b b ( c d ). Zero and Negative Integer Eponents Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero. ( y ) ( y z ) 6 a a. Zero and Negative Integer Eponents
12 Eamples: Simplify each of the following; epress all answers so that eponents are positive. Whenever an eponent is 0 or negative, we assume the base is not zero. r r r r 7 6y 1 y. Zero and Negative Integer Eponents Section. Review Zero eponents Negative integer eponents Variable eponents Evaluated and simplified epressions Section. Learning Objectives Scientific notation Writing numbers in scientific notation Changing from scientific notation to standard notation Using scientific notation to simplify computations
13 Scientific Notation Scientific notation is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers. Ordinary decimal notation Scientific notation 00 10,000 10,70,000, Eamples: Writing numbers in scientific notation Write in scientific notation: 1) The sun is appro. 9,000,000 miles away ) Changing from scientific notation to standard notation
14 Summary Here are some more eamples of scientific notation = = = = = = = = (not usually done) 1 = /10 = 0.1 = = (not usually done) 1/100 = 0.01 = =. 10-1/1000 = = = /10000 = = = Using Scientific Notation to Simplify Computations 1) As of December 007, currency in circulation that is, U.S. coins and paper currency in the hands of the public totaled about $89 billion dollars. What is the scientific notation of the currency? $ ) The deficit for fiscal year 009, which ended Sept. 0, came in at a record $ What is the standard notation of the deficit? $1. trillion Section. Review Scientific notation Writing numbers in scientific notation Changing from scientific notation to standard notation Using scientific notation to simplify computations
15 Homework Assignment Read pages Homework: pages #17, 19,, 7-7 odds,,, -67 odds, 7-97 odds, 101, 10, 109, 111 Read pages Homework: pages 9-97 #19-9 odds, 9-67 odds, 71, 7, 79, 8, 8, 87, 89, 91, 101
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