University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

188

Pages:

14
Author:

Polizistuff

§ 12.2 Negative Exponents and Scientific Notation Negative Exponents Using the quotient rule from section 3.1, 4 x 4−6 −2 =x =x 6 x x≠0 But what does x -2 mean? 4 x x⋅ x⋅ x⋅ x 1 1 = = = 2 6 x x⋅ x⋅ x⋅ x⋅ x⋅ x x⋅x x 15 Negative Exponents So, in order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows. If a ≠ 0, and n is an integer, then a −n 1 = n a 16 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents or calculating. 1 1 3 = 2 = 3 9 1 −7 x = 7 x −2 2 2x = 4 x −4 Remember that without parentheses, x is the base for the exponent –4, not 2x 17 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents or calculating. −x −3 1 = − 3 x 1 1 −3 = − 2 = − 3 9 −2 (−3) −2 = 1 1 = 2 (−3) 9 Notice the difference in results when the parentheses are included around −3. 18 Simplifying Expressions Example Simplify by writing each of the following expressions with positive exponents. 1 x 1 3 = = =x −3 1 x 1 3 x 3 1) −2 2) x y −4 1 4 2 y x = = 2 1 x 4 y (Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from a numerator to a denominator, or vice versa, and switch the exponent to its positive value.) 19 Summary of Exponent Rules If m and n are integers and a and b are real numbers, then: Product Rule for exponents am · an = am+n Power Rule for exponents (am)n = amn Power of a Product (ab)n = an · bn n an a Power of a Quotient   = n , b ≠ 0 b b Quotient Rule for exponents am m−n = a , a≠0 n a Zero exponent a0 = 1, a ≠ 0 Negative exponent a −n = 1 , a≠0 n a 20 Simplifying Expressions Simplify by writing the following expression with positive exponents or calculating. −2  3 ab   − 4 7 −3  = (3 a b ) − 2 = (3 −)2 (a −)2 (b ) − 2 3 a b  (3−4 a 7b −3 ) (3−4 ) (a 7 ) (b −3 ) −2 3 −2 −2 3 −2 Power of a quotient rule − 2 −2 3 −2 Power of a product rule 8 8 34 a −6b 2 34 a14b 2 a a = 8 −14 6 = 8 6 6 = 34−8 a14−6b 2−6 = 3−4 a 8b −4 = 4 4 = 3b 3a b 3ab 81b 4 Power rule for exponents Quotient rule for exponents Negative exponents Negative exponents 21 Scientific Notation In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers. A positive number is written in scientific notation if it is written as a product of a number a, where 1 ≤ a < 10, and an integer power r of 10. a × 10r 22 Scientific Notation To Write a Number in Scientific Notation 1) Move the decimal point in the original number to the left or right, so that the new number has a value between 1 and 10. 2) Count the number of decimal places the decimal point is moved in Step 1. • If the original number is 10 or greater, the count is positive. • If the original number is less than 1, the count is negative. 3) Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. 23 Scientific Notation Example Write each of the following in scientific notation. Have to move the decimal 3 places to the left, so that the 1) 4700 new number has a value between 1 and 10. Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3. 4700 = 4.7 × 103 2) 0.00047 Have to move the decimal 4 places to the right, so that the new number has a value between 1 and 10. Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4. 0.00047 = 4.7 × 10-4 24 Scientific Notation To Write a Scientific Notation Number in Standard Form • Move the decimal point the same number of spaces as the exponent on 10. • If the exponent is positive, move the decimal point to the right. • If the exponent is negative, move the decimal point to the left. 25 Scientific Notation Example Write each of the following in standard notation. 1) 5.2738 × 103 Since the exponent is a positive 3, we move the decimal 3 places to the right. 5.2738 × 103 = 5273.8 2) 6.45 × 10-5 Since the exponent is a negative 5, we move the decimal 5 places to the left. 00006.45 × 10-5 = 0.0000645 26 Operations with Scientific Notation Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Example Perform the following operations. 1) (7.3 × 10-2)(8.1 × 105) = (7.3 · 8.1) × (10-2 · 105) = 59.13 × 103 = 59,130 1.2 ×10 4 1.2 10 4 −5 = 0 . 3 × 10 = × = 0.000003 2) 9 9 4 ×10 4 10 27