Integer Exponents and Scientific Notation 5.1 Integer Exponents and Scientific Notation 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.

Integer Exponents and Scientific Notation We use exponents to write products of repeated factors. For example, 25 is defined as 2 • 2 • 2 • 2 • 2 = 32. The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 25 is called an exponential or a power. We read 25 as “2 to the fifth power” or “2 to the fifth.” Slide 5.1- 2

Use the product rule for exponents. Objective 1 Use the product rule for exponents. Slide 5.1- 3

Product Rule for Exponents Use the product rule for exponents. Product Rule for Exponents If m and n are natural numbers and a is any real number, then am • an = am + 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 5.1- 4

m8 • m6 m5 • p4 (–5p4) (–9p5) (–3x2y3) (7xy4) CLASSROOM EXAMPLE 1 Using the Product Rule for Exponents Apply the product rule, if possible, in each case. Solution: m8 • m6 m5 • p4 (–5p4) (–9p5) (–3x2y3) (7xy4) Slide 5.1- 5

Define 0 and negative exponents. Objective 2 Define 0 and negative exponents. Slide 5.1- 6

Zero Exponent If a is any nonzero real number, then a0 = 1. Define 0 and negative exponents. Zero Exponent If a is any nonzero real number, then a0 = 1. The expression 00 is undefined. Slide 5.1- 7

290 (–29)0 –290 80 – 150 Using 0 as an Exponent Evaluate. CLASSROOM EXAMPLE 2 Using 0 as an Exponent Evaluate. 290 (–29)0 –290 80 – 150 Slide 5.1- 8

Define 0 and negative exponents. For any natural number n and any nonzero real number a, A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Slide 5.1- 9

6-5 (2x)-4 –7p-4 Using Negative Exponents CLASSROOM EXAMPLE 3 Using Negative Exponents Write with only positive exponents. 6-5 (2x)-4 –7p-4 Evaluate: 4-1 – 2-1 Slide 5.1- 10

Using Negative Exponents CLASSROOM EXAMPLE 4 Using Negative Exponents Evaluate. Slide 5.1- 11

Special Rules for Negative Exponents Define 0 and negative exponents. Special Rules for Negative Exponents If a ≠ 0 and b ≠ 0 , then and Slide 5.1- 12

Use the quotient rule for exponents. Objective 3 Use the quotient rule for exponents. Slide 5.1- 13

Quotient Rule for Exponents Use the quotient rule for exponents. Quotient Rule for Exponents If a is any nonzero real number and m and n are integers, then 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 5.1- 14

Using the Quotient Rule for Exponents CLASSROOM EXAMPLE 5 Using the Quotient Rule for Exponents Apply the quotient rule, if possible, and write each result with only positive exponents. Slide 5.1- 15

Use the power rules for exponents. Objective 4 Use the power rules for exponents. Slide 5.1- 16

Power Rule for Exponents Use the power rules for exponents. Power Rule for Exponents If a and b are real numbers and m and n are integers, then and That is, a) To raise a power to a power, multiply exponents. 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. Slide 5.1- 17

Using the Power Rules for Exponents CLASSROOM EXAMPLE 6 Using the Power Rules for Exponents Simplify, using the power rules. Solution: Slide 5.1- 18

Special Rules for Negative Exponents, Continued Use the power rules for exponents. Special Rules for Negative Exponents, Continued If a ≠ 0 and b ≠ 0 and n is an integer, then 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. Slide 5.1- 19

Using Negative Exponents with Fractions CLASSROOM EXAMPLE 7 Using Negative Exponents with Fractions Write with only positive exponents and then evaluate. Solution: Slide 5.1- 20

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 Slide 5.1- 21

Use the power rules for exponents. Definition and Rules for Exponents, Continued Negative Exponent Power Rules Special Rules Slide 5.1- 22

Simplify exponential expressions. Objective 5 Simplify exponential expressions. Slide 5.1- 23

Using the Definitions and Rules for Exponents CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents Simplify. Assume that all variables represent nonzero real numbers. Solution: Slide 5.1- 24

Simplify. Assume that all variables represent nonzero real numbers. CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents (cont’d) Simplify. Assume that all variables represent nonzero real numbers. Slide 5.1- 25

Use the rules for exponents with scientific notation. Objective 6 Use the rules for exponents with scientific notation. Slide 5.1- 26

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 10. This is often a simpler way to express very large or very small numbers. Slide 5.1- 27

Use the rules for exponents with scientific notation. A number is written in scientific notation when it is expressed in the form a × 10n where 1 ≤ |a| < 10 and n is an integer. Slide 5.1- 28

Converting to Scientific Notation Use the rules for exponents with scientific notation. Converting to Scientific Notation Step 1 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 10. Step 3 Determine the sign for the exponent. Decide whether multiplying by 10n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less. Slide 5.1- 29

CLASSROOM EXAMPLE 9 Writing Numbers in Scientific Notation Write the number in scientific notation. 29,800,000 346,100,000,000 102,000,000,000,000 Slide 5.1- 30

CLASSROOM EXAMPLE 9 Writing Numbers in Scientific Notation (cont’d) Write the number in scientific notation. 0.0000000503 0.000000000385 0.00000000004088 Slide 5.1- 31

Converting a Positive Number from Scientific Notation Use the rules for exponents with scientific notation. Converting a Positive Number from Scientific Notation Multiplying a positive number by a positive power of 10 makes the number greater, so move the decimal point to the right if n is positive in 10n. Multiplying a positive number by a negative power of 10 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 5.1- 32

CLASSROOM EXAMPLE 10 Converting from Scientific Notation to Standard Notation Write each number in standard notation. 2.51 ×103 5.07 ×10−7 –6.8 ×10−4 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 5.1- 33

Using Scientific Notation in Computation CLASSROOM EXAMPLE 11 Using Scientific Notation in Computation Evaluate Slide 5.1- 34

CLASSROOM EXAMPLE 12 Using Scientific Notation to Solve Problems The distance to the sun is 9.3 × 107 mi. How long would it take a rocket traveling at 3.2 × 103 mph to reach the sun? Slide 5.1- 35