5.1. Integer Exponents and Scientific Notation. Objectives. Use the product rule for exponents. Define 0 and negative exponents.

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1 Chapter 5 Section

2 5. Integer Exponents and Scientific Notation Objectives Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. Use the power rules for exponents. Simplify exponential expressions. Use the rules for exponents with scientific notation.

3 Integer Exponents and Scientific Notation We use exponents to write products of repeated factors. For example, 2 5 is defined as The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 2 5 is called an exponential or a power. We read 2 5 as 2 to the fifth power or 2 to the fifth. Slide 5.-

4 Objective Use the product rule for exponents. Slide 5.- 4

5 Use the product rule for exponents. Product Rule for Exponents If m and n are natural numbers and a is any real number, then a m a n a m + n. That is, when multiplying powers of like bases, keep the same base and add the exponents. Be careful not to multiply the bases. Keep the same base and add the exponents. Slide 5.- 5

6 CLASSROOM EXAMPLE Using the Product Rule for Exponents Apply the product rule, if possible, in each case. Solution: m 8 m 6 m 8+6 m 4 m 5 p 4 Cannot be simplified further because the bases m and p are not the same. The product rule does not apply. ( 5p 4 ) ( 9p 5 ) ( 5)( 9)(p 4 p 5 ) 45p p 9 ( x 2 y ) (7xy 4 ) ( )(7) x 2 xy y 4 2x 2+ y +4 2x y 7 Slide 5.- 6

7 Objective 2 Define 0 and negative exponents. Slide 5.- 7

8 Define 0 and negative exponents. Zero Exponent If a is any nonzero real number, then a 0. The expression 0 0 is undefined. Slide 5.- 8

9 CLASSROOM EXAMPLE 2 Evaluate. Using 0 as an Exponent Solution: 29 0 ( 29) (29 0 ) Slide 5.- 9

10 Define 0 and negative exponents. Negative Exponent For any natural number n and any nonzero real number a, a n. n a A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. 9 2 Not negative Negati 2 ve Slide 5.- 0

11 CLASSROOM EXAMPLE Write with only positive exponents. Solution: Using Negative Exponents (2x) -4, x 0, x 0 ( 2x) 4 7p -4, p 0 7 p 4 7 4, p 0 p Evaluate Slide 5.-

12 CLASSROOM EXAMPLE 4 Evaluate. Using Negative Exponents Solution: Slide 5.- 2

13 Define 0 and negative exponents. Special Rules for Negative Exponents If a 0 and b 0, then a a n n and n m a b. m n b a Slide 5.-

14 Objective Use the quotient rule for exponents. Slide 5.- 4

15 Use the quotient rule for exponents. Quotient Rule for Exponents If a is any nonzero real number and m and n are integers, then a a m n m n a. That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator. Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses. Slide 5.- 5

16 CLASSROOM EXAMPLE 5 Apply the quotient rule, if possible, and write each result with only positive exponents. Solution: Using the Quotient Rule for Exponents m m x, y 0 5 y 8 5 m m 5, m 0 m 6 ( 8) , or 25 Cannot be simplified because the bases x and y are different. The quotient rule does not apply. Slide 5.- 6

17 Objective 4 Use the power rules for exponents. Slide 5.- 7

18 Use the power rules for exponents. Power Rule for Exponents If a and b are real numbers and m and n are integers, then That is, and ( ) n ( ) m mn m m a) a a, b) ab a b, m a a c) ( b 0). m b b a) To raise a power to a power, multiply exponents. m b) To raise a product to a power, raise each factor to that power. c) To raise a quotient to a power, raise the numerator and the denominator to that power. m Slide 5.- 8

19 CLASSROOM EXAMPLE 6 Simplify, using the power rules. Solution: Using the Power Rules for Exponents ( 5 r ) 4 ( 5 r ) r 20 r ( 5 y ) 2 ( ) 2 ( 5 ) 2 y y 9y Slide 5.- 9

20 Use the power rules for exponents. Special Rules for Negative Exponents, Continued If a 0 and b 0 and n is an integer, then a n a n and That is, any nonzero number raised to the negative nth power is equal to the reciprocal of that number raised to the nth power. n n a b b a. Slide

21 CLASSROOM EXAMPLE 7 Using Negative Exponents with Fractions Write with only positive exponents and then evaluate. Solution: x 5, x 0 2x 5 5 2x 5 2x Slide 5.- 2

22 Use the power rules for exponents. Definition and Rules for Exponents For all integers m and n and all real numbers a and b, the following rules apply. Product Rule Quotient Rule Zero Exponent a a a m n m+ n a a a m n 0 ( a 0) m n a ( a ) 0 Slide

23 Use the power rules for exponents. Definition and Rules for Exponents, Continued Negative Exponent a n ( a 0) n a Power Rules Special Rules a n n ( m a ) a b a m mn ( ab) m a m b n a a 0 n a n a ( a 0) ( ) m m m ( b 0) a b n a b n m a b m n b a n ( a, b 0) b a ( a, b 0) Slide 5.- 2

24 Objective 5 Simplify exponential expressions. Slide

25 CLASSROOM EXAMPLE 8 Simplify. Assume that all variables represent nonzero real numbers. Solution: ( 2 4 ) 5 x x x ( 2 ) m n 2 m n 2( 5) 4 Using the Definitions and Rules for Exponents ( ) x + + ( ) m m n n 4 x x 2 2 m n 4 2 m n m n m n 4 2 m n 4 ( ) 2 m n 4+ m n mn Slide

26 CLASSROOM EXAMPLE 8 2 2y 4y x x Using the Definitions and Rules for Exponents (cont d) Simplify. Assume that all variables represent nonzero real numbers. Solution: 2 y 4 x x y Combination of rules 2 4 y 5 x y 5 4x y 5 x Slide

27 Objective 6 Use the rules for exponents with scientific notation. Slide

28 Use the rules for exponents with scientific notation. In scientific notation, a number is written with the decimal point after the first nonzero digit and multiplied by a power of 0. This is often a simpler way to express very large or very small numbers. Slide

29 Use the rules for exponents with scientific notation. Scientific Notation A number is written in scientific notation when it is expressed in the form a 0 n where a < 0 and n is an integer. Slide

30 Use the rules for exponents with scientific notation. Converting to Scientific Notation Step Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed. Step 2 Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 0. Step Determine the sign for the exponent. Decide whether multiplying by 0 n should make the result of Step greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less. Slide 5.- 0

31 CLASSROOM EXAMPLE 9 Write the number in scientific notation. 29,800,000 ^ Solution: Writing Numbers in Scientific Notation Step Place a caret to the right of the 2 (the first nonzero digit) to mark the new location of the decimal point. Step 2 Count from the decimal point 7 places, which is understood to be after the caret. 29,800, ,800,000. Decimal point moves 7 places to the left Step Since 2.98 is to be made greater, the exponent on 0 is positive. 29,800, Slide 5.-

32 CLASSROOM EXAMPLE ^ Writing Numbers in Scientific Notation (cont d) Write the number in scientific notation. Solution: Step Place a caret to the right of the 5 (the first nonzero digit) to mark the new location of the decimal point. Step 2 Count from the decimal point 8 places, which is understood to be after the caret Decimal point moves 7 places to the left Step Since 5.0 is to be made less, the exponent 0 is negative Slide 5.- 2

33 Use the rules for exponents with scientific notation. Converting a Positive Number from Scientific Notation Multiplying a positive number by a positive power of 0 makes the number greater, so move the decimal point to the right if n is positive in 0 n. Multiplying a positive number by a negative power of 0 makes the number less, so move the decimal point to the left if n is negative. If n is 0, leave the decimal point where it is. Slide 5.-

34 CLASSROOM EXAMPLE 0 Converting from Scientific Notation to Standard Notation Write each number in standard notation. Solution: Move the decimal places to the right Move the decimal 4 places to the left. When converting from scientific notation to standard notation, use the exponent to determine the number of places and the direction in which to move the decimal point. Slide 5.- 4

35 CLASSROOM EXAMPLE Evaluate 200, Solution: Using Scientific Notation in Computation Slide 5.- 5

36 CLASSROOM EXAMPLE 2 Using Scientific Notation to Solve Problems The distance to the sun is mi. How long would it take a rocket traveling at.2 0 mph to reach the sun? Solution: d rt, so d t r It would take approximately hours. Slide 5.- 6