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Chapter 7: Exponents and Polynomials Integer Exponents Notes 7.1 Integers: _____________________________________________________________________________ ___________________________________________ Exponents: ___________________________________________________________________________ _____________________________________________________________________________________ When you have a __________________________, youβre talking about a very small value. You are often asked to express your answer with no negative exponents. To do this, flip the negative exponent to the other half of the fraction and make it positive. Express the following with only positive exponents: a.) πβ5 b.) π2 πβ3 πβ1 c.) 3β4 d.) π₯ βπ π₯ β3 e.) π0 π¦5 Zero Exponents Anything to the zero power is __________, but only what is being raised to the zero power. Examples: a.) 40 b.) β20 c.) (β6)0 d.) 5π0 π π0 e.) π3 Evaluate the following: 1.) πβ3 for p = 4 2.) 8πβ2 π0 for a = -2, and b = 6 3.) 5π β3 π β6 for r = 3 and s = 1 1 Chapter 7: Exponents and Polynomials Powers of 10 and Scientific Notation Notes 7.2 The table below shows the relationships between several powers of ten: Power 103 102 101 100 10β1 10β2 10β3 Value Each time you divide by 10, the exponent decreases by 1 and the decimal point moves 1 place to the left (negative exponents). Likewise, each time you multiply by 10, the exponent increases by 1 and the decimal point moves 1 place to the right (positive exponents). Powers of 10 Words Numbers Positive Integer Exponent: If n is a positive integer, find the value of 10π by starting with one and moving the decimal point n places to the right. Negative Integer Exponent: If n is a negative integer, find the value of 10βπ by starting with one and moving the decimal point n places to the left. Scientific Notation: consists of two parts: (1 < n < 9) and 10 raised to a power. Standard Form: Some know this form as βexpanded formβ. Fill in the following: Scientific Notation Standard Form 1.2 x 103 50,400,000,000 8.03 x 10β7 0.0000000309 5.2 x 105 120 2 Chapter 7: Exponents and Polynomials Fill in your own examples, two with positive exponents, and two with negative exponents: Scientific Notation Standard Form Comparing and ordering numbers in Scientific Notation: Order the list of numbers from least to greatest: 1.) 1.2 x 10β1, 8.2 x 104 , 6.2 x 105 , 2.4 x 105 , 1 x 10β1, 9.9 x 10β4 2.) 5.2 x 10β3, 3 x 1014 , 4 x 10β3, 2 x 10β12, 4.5 x 1030, 4.5 x 1014 3 Chapter 7: Exponents and Polynomials Multiplication Properties of Exponents Notes 7.3 You have seen that using exponents are helpful when writing very small and very large numbers. To perform operations on these numbers, you can use properties of exponents. You can also use these properties to simplify expressions. When an expression is fully simplified, follow this checklist: οΌ There are no negative exponents οΌ The same base doesnβt occur more than once in the product or quotient οΌ No powers, products or quotients are raised to powers οΌ Numerical coefficients in both numerator and denominator have no factors in common. When multiplying: Coefficients: Base: Exponents: 1.) 24 β’ 23 2.) π¦ β5 β’ π¦ 3 β’ π₯ 8 π¦ β4 β’ π¦ 2 3.) π β’ πβ3 β’ π4 4 Chapter 7: Exponents and Polynomials When raising a power to a power: Coefficients: Base: Exponents: 4.) (74 )3 5.) (36 )0 6.) (π₯ 2 )β4 π₯ 5 8.) β(3π₯ 3 )2 9.) (π₯ β2 π¦ 0 )3 When finding the powers of productsβ¦ Simplify: 7.) (β3π₯ 5 )4 5 Chapter 7: Exponents and Polynomials Simplify the following powers of products: 10.) (5π₯ 6 π¦ 2 )( 2π₯ 2 π¦ 4 )3 11.) (4π)5 12.) (π₯ 2 π¦ 3 )4 ( π₯ 2 π¦ 4 )β4 Applying: 13.) Light from the sun travels at about 1.86 x 105 miles per second. It takes about 500 seconds for the light to reach Earth. Find the approximate distance from the Sun to Earth. Write your answer in scientific notation. 14.) Light travels at 1.86 x 105 miles per second. Find the approximate distance that light travels in one hour. Write your answer in scientific notation. 6 Chapter 7: Exponents and Polynomials Division Properties of Exponents Notes 7.4 The quotient of powers with the same base can be found by writing the powers in factored form and dividing out common factors. Remember in the last section when we multiply with the same base, you add the exponents. So keeping that in mind, when we divide, we will need to subtract the exponents. *HOWEVER, it is crucial to keep in mind where you start because thatβs where you end. If you start your subtraction in the numerator, your answer will go in the numerator. If you start your subtracting in the denominator, your answer goes in the denominator. Remember to be fully simplified; you need to have only positive exponents. When dividing with exponents: Coefficients: Base: Exponents: Simplify the following: 1.) 38 32 2.) π5 π9 (ππ)4 π5 π4 (π5 )2 π 3.) 7 Chapter 7: Exponents and Polynomials Simplify the following: 3 3 4 4.) ( ) π βπ 7.) (π ) 3 2π₯ 3 ) π¦π§ 5.) ( 2 β3 8.) (5) 3 π3 π ) π2 π2 6.) ( 3 β1 9.) (4) 2π₯ β2 (3π¦) Exception: Only with scientific notation can the exponent be negative in your answer. 10.) (2 x 108 ) ÷ (8 x 105 ) 11.) (3.3 x 106 ) ÷ (3 x 108 ) 12.) In the year 2000, the United States public debt was about 5.6 x 1012 dollars. The population of the United States in that year was about 2.8 x 108 people. What was the average debt per person? Give your answer in standard form. 8 Chapter 7: Exponents and Polynomials Polynomials Notes 7.5 Monomial: ___________________________________________________________________________ _____________________________________________________________________________________ Degree of Monomial: ___________________________________________________________________ _____________________________________________________________________________________ State the degree of the following Monomials: 1.) 5 2.) -5π₯ 1 3.) 6π5 π 2 4.) -2π2 π 5.) 4π₯ 0 Polynomial: ___________________________________________________________________________ _____________________________________________________________________________________ Degree of a Polynomial: _________________________________________________________________ _____________________________________________________________________________________ Find the degree of the following: 6.) 4x - 18π₯ 5 7.) .5π₯ 2 π¦ + .25xy + .75 Standard form of a Polynomial: __________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Leading Coefficient: ____________________________________________________________________ _____________________________________________________________________________________ 9 Chapter 7: Exponents and Polynomials Names for Polynomials: First name, Last name = Degree name, Term name Degree Name Terms 0 1 1 2 2 3 3 4 or more Name 4 5 6 or more Name the following: 8.) π₯ 3 + π₯ 2 - x + 2 9.) 6 10.) -3π¦ 8 + 18π₯ 2 π¦ 3 + 14y 11.) 4π₯ 2 y + 8x + 2π₯ 2 yπ§ 2 - 5π₯ 3 π§ 0 β 7 12.) A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a five second fuse. The height of the firework in feet is given by the polynomial: -16π‘ 2 + 200t + 6 where t is the time in seconds. How high will the firework be when it explodes? 10 Chapter 7: Exponents and Polynomials Adding and Subtracting Polynomials Notes 7.6 Like terms must have two things in common: 1.) Same __________________ 2.) Same __________________ Rules for adding and subtracting with exponents: 1.) The base stays the same 2.) The exponents stay the same 3.) Apply the math to the coefficients *Always express your answer in standard form! Two methods: Vertical method: Line up the problem so that the like terms are on top of each other and the operation can be performed vertically. Remember, with subtraction, first do keep change change. 1.) (15π3 + 6π2 ) + 2π3 2.) (-4π3 - 13π2 + 2g) β (-5π2 + 8g β 9) Horizontal method: Line up the problem in one horizontal row by grouping the like terms together to simplify. 3.) (π4 - 2a) β (3π4 - 3a + 1) 4.) (3π₯ 2 - 2x + 8) + (x - π₯ 2 - 4) 11