1. § 5.5 Negative Exponents and Scientific Notation

2. Martin-Gay, Beginning and Intermediate Algebra, 4ed 22 Negative Exponents If a is a real number other than 0 and n is an integer, then Example: Simplify by writing each expression with positive exponents. Remember that without parentheses, x is the base for the exponent –4, not 2x

3. Martin-Gay, Beginning and Intermediate Algebra, 4ed 33 Simplify by writing each of the following expressions with positive exponents. Simplifying Expressions Notice the difference in results when the parentheses are included around  3. Example:

4. Martin-Gay, Beginning and Intermediate Algebra, 4ed 44 Simplify by writing each of the following expressions with positive exponents. (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.) Simplifying Expressions Example:

5. Martin-Gay, Beginning and Intermediate Algebra, 4ed 55 Power of a quotient rulePower of a product rule Power rule for exponents Quotient and product rules for exponents Negative exponents Negative exponent Simplifying Expressions Simplify by writing each of the following expressions with positive exponents. Example:

6. Martin-Gay, Beginning and Intermediate Algebra, 4ed 66 If m and n are integers and a and b are real numbers, then: Product Rule for exponents a m · a n = a m+n Power Rule for exponents (a m ) n = a mn Power of a Product (ab) n = a n · b n Power of a Quotient Quotient Rule for exponents Zero exponent a 0 = 1, a  0 Negative exponent Summary of Exponent Rules

7. Martin-Gay, Beginning and Intermediate Algebra, 4ed 77 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  10 r Scientific Notation

8. Martin-Gay, Beginning and Intermediate Algebra, 4ed 88 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. Scientific Notation

9. Martin-Gay, Beginning and Intermediate Algebra, 4ed 99 Write each of the following in scientific notation. 4700 1) Have to move the decimal 3 places to the left, so that the 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  10 3 0.00047 2) 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 Scientific Notation Example:

10. Martin-Gay, Beginning and Intermediate Algebra, 4ed 10 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. Scientific Notation

11. Martin-Gay, Beginning and Intermediate Algebra, 4ed 11 Write each of the following in standard notation. 5.2738  10 3 1) Since the exponent is a positive 3, we move the decimal 3 places to the right. 5.2738  10 3 = 5273.8 6.45  10 -5 2) Since the exponent is a negative 5, we move the decimal 5 places to the left. 00006.45  10 -5 = 0.0000645 Scientific Notation Example:

12. Martin-Gay, Beginning and Intermediate Algebra, 4ed 12 Operations with Scientific Notation Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Perform the following operations. = (7.3 · 8.1)  (10 -2 · 10 5 ) = 59.13  10 3 = 59,130 (7.3  10 -2 )(8.1  10 5 ) 1) 2) Example: