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Algebra I Chapter 7 Notes Rules of Exponents Section 7-1 Monomial – Constant – Base – Exponent - Section 7-1 Monomial – a number, a variable, or the product of a number and variable with non-negative, integer exponents Constant – a monomial that is a real number Base – term being multiplied in an exponential expression Exponent – the number of times the base is multiplied in an exponential expression Section 7-1: Multiplication Rules of Exponents, Day 1 Ex) Determine whether each expression is a monomial. Write yes or no, explain WHY. a) 10 b) f + 24 c) x -5 d) h 2 e) -5y Section 7-1: Multiplication Rules of Exponents, Day 1 Product of Powers Words To multiply two powers that have the same base, you add their exponents Symbols For any real number a, and any integers m and p: Examples x3 × x 5 = am × a p = a m+p 2 ×2 = Ex) Simplify each expression a) (6n3 )(2n 7 ) b) (3pt 3 )(p3t 4 ) 7 10 c) (-4rx 2t 3 )(-6r 5 x 2t) Section 7-1: Multiplication Rules of Exponents, Day 1 Power of a Power Words To raise a power to another power, you multiply the exponents Symbols For any real number a, and any integers m and p, Example (x 3 )5 = Ex) Simplify a) [(23 )2 ]4 (2 7 )10 = b) [(2 2 )2 ]4 (a m ) p = a m×p Section 7-1: Multiplication Rules of Exponents, Day 2 Power of a Product Words Exponents can be distributed when terms are being multiplied Symbols For any real numbers a and b, and any integer m, Examples (-2xy3 )5 = (ab)m = a m bm Ex) Simplify each expression a) (xy 4 )6 b) (4a 4b9c)2 c) (-2 f 2 g3h2 )3 d) (-3p5t 6 )4 Section 7-1: Multiplication Rules of Exponents, Day 2 Ex) Use all rules to simplify a) (3xy 4 )2 [(-2y)2 ]3 b) (-7ab4c)3[(2a2c)2 ]3 c) (5x 2 y)2 (2xy3z)3 (4yxz) d) (-2g3h)(-3gj 4 )2 (-ghj)2 Section 7-2: Division Rules of Exponents, Day 1 Quotient of Powers Words To divide two powers with the same base, subtract the exponents Symbols For any nonzero number a, and any integers m and p, Examples x11 = 8 x Ex. Simplify g 3h 5 a) gh 2 = am = a m-p p a 520 = 7 5 b) x3y4 x2 y c) k 7 m10 p = 5 3 km p Section 7-2: Division Rules of Exponents, Day 1 Power of a Quotient Words To find the power of a quotient, find the power of the numerator and the power of the denominator m a a m For any real numbers a and b does not = zero, ( ) = and any integer m, b bm Symbols r ( )5 = t 3 ( )4 = 5 Examples Ex) Simplify 3 3p a) ( )2 = b) 7 3x 4 3 ( ) = 4 c) 2y 2 2 ( 3) = 3z 3 4x d) ( 4 )3 = 5y Section 7-2: Division Rules of Exponents, Day 1 Simplify using division rules of exponents 4 2 12 3 m p p 1) 2) t r 3) c 4 d 4 f 3 4) 2 m p 2 p tr c2 d 4 f 3 3xy 4 2 ( 2 ) 5z 2r 3t 6 4 ( ) 3 4rt 3m 5r 3 3 ( ) 2 6m r 5) 6) Section 7-2: Division Rules of Exponents, Day 2 Zero Exponent Property Words Any nonzero number raised to the zero power is equal to 1 Symbols For any nonzero number a, an0p =1 -5 Example 4 b r0 = ( ) = c 15 = -2 0 Ex) Simplify. Assume no denominator = zero 2 5 2 4 2 0 5 0 4n q r b 0 a) (- 3 2 ) b) x y = c) c d = 9n q r x 3 b2c Section 7-2: Division Rules of Exponents, Day 2 Negative Exponent Property Words For any nonzero number a, and any integer n, Symbols For any nonzero number a and integer n, Examples 1 1 2 = 4= 2 16 -4 or 1 = -4 j a a-n = 3m 5r 3 3 ) 46m 2 r j ( -nis the reciprocal of 1 an Ex) Simplify. NO NEGATIVE EXPONENTS! -5 4 -3 4 2 3 -5 -3 2 n p 5r t 2a b c v wx a) -2 = b) = c) = d) = 2 7 -5 -3 -1 -4 r -20r t u 10a b c wy-6 1 an Section 7-2: Division Rules of Exponents, Day 2 Simplify using division rules of exponents 1) (4k m ) 2) 20qr t 3 2 3 (5k 2 m-3 )-2 3) -3x -6 y -1z -2 -2 ( ) -2 -5 6x yz -2 -5 4q 0 r 4t -2 -2 4 2 2a b c -1 4) ( ) -2 -5 -7 -4a b c Section 7-4: Scientific Notation Scientific Notation – a number written in the form a ´10n, where 1 < a < 10 and n is an integer. Ex) Write the following numbers in scientific notation. 1) 201,000,000 2) 0.000051 Section 7-4: Scientific Notation Ex) Write the following numbers in standard form 1) 6.3´10 9 2) 4 ´10-7 Section 7-4: Multiplying with Scientific Notation Ex) Use rules of exponents to multiply the following numbers together. Express your answer in both scientific notation and standard form! 1) (3.5´10-3 )(7´105 ) 2) (7.8´10-4 )2 Section 7-4: Dividing with Scientific Notation Ex) Use rules of exponents to multiply the following numbers together. Express your answer in both scientific notation and standard form! 1) 3.066 ´10 2) 1.305´10 8 7.3´10 3 3 1.45´10-4 Section 7-5: Exponential Functions – Exponential Growth, Day 1 Exponential Function – A function that can be x written in the form y = ab , where a cannot be 0, b > 0, and b cannot be 1. Examples of exponential 1 x x x functions: y = 2(3) , y = 4 , or y = ( 2 ) Exponential Growth Equation Domain and Range Intercepts End Behavior f (x) = ab x where a > 0, b >1 Section 7-5: Exponential Functions – Exponential Growth, Day 1 Graph of Exponential Growth Section 7-5: Exponential Functions – Exponential Growth, Day 1 Ex) Graph y = 3x, Find the y-intercept, and state the domain and range. You will have to create a table to graph! What is the pattern on the table? x -2 -1 0 1 2 3x y Section 7-5: Exponential Functions – Exponential Decay, Day 2 Exponential Decay Equation Domain and Range Intercepts End Behavior f (x) = ab x where a > 0, 0 < b <1 Section 7-5: Exponential Functions – Exponential Decay, Day 2 Graph of Exponential Decay Section 7-5: Exponential Functions – Exponential Decay, Day 2 Ex) Graph y = (13) , Find the y-intercept, and state the domain and range. You will have to create a table to graph! What is the pattern on the table? x x -2 -1 0 1 2 1 ( )x 3 y Section 7-6: Exponential Growth and Decay Patterns, Day 1 Equation for Exponential Growth: y = a(1+ r)t a: initial amount t : time y: final amount r: rate of change expressed as a decimal, r > 0 Ex) The prize for a radio station contest begins with a $100 gift card. Once a day, a name is announced. The person has 15 minutes to call or the prize increases 2.5% for the next day. a) Write an equation representing the amount of the gift card after t days with no winner b) How much will the card be worth if no one claims it after 10 days? Section 7-6: Exponential Growth and Decay Patterns, Day 1 Compound Interest – interest earned or paid on both the initial investment AND previously earned interest. It is an application of exponential growth. Equation for Compound Interest r nt A = P(1+ ) n A: current amount P: principal/initial amount r: annual interest rate expressed as a decimal n: number of times interest is compounded per year t: time in years Section 7-6: Exponential Growth and Decay Patterns, Day 1 Ex) Maria’s parents invested $14,000 at 6% per year compounded monthly. How much money will there be in the account after 10 years? Section 7-6: Exponential Growth and Decay Patterns, Day 2 Equation for Exponential Decay y = a(1- r) t a: initial amount y = final amount t: time r: rate of decay as a decimal 0 < r < 1 Ex) A fully inflated raft is losing 6.6% of its air every day. The raft originally contained 4500 cubic inches of air. a) Write an equation representing the loss of air b) Estimate the amount of air in the raft after 7 days Section 7-6: Exponential Growth and Decay Patterns, Day 2 Solve the 3 problems. You must choose which equation to use on each. 1) Paul invested $400 into an account with 5.5% interest compounded monthly. How much will he have in 8 years? 1) Ms. Acosta received a job as a teacher with a starting salary of $34,000. She will get a 1.5% increase in her salary each year. How much will she earn in 7 years? 1) In 2000 the 2200 students attended East High School. The enrollment has been declining 2% annually. If this continues, how many students will be enrolled in the year 2015?