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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.2 Negative Exponents and Scientific Notation

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Negative Exponents Using the quotient rule from section 3.1, But what does x -2 mean?

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 is a real number other than 0, and n is an integer, then Negative Exponents

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplifying Expressions Simplify. Write each result using positive exponents only. Example Don’t forget that since there are no parentheses, x is the base for the exponent –4. Helpful Hint

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify. Write each result using positive exponents only. Simplifying Expressions Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify by writing each of the following expressions with positive exponents. 1) 2) (Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from the numerator to the denominator, or vice versa, and switch the exponent to its opposite value.) Simplifying Expressions Example

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Simplify by writing the following expression with positive exponents or calculating. Power of a quotient rulePower of a product rule Power rule for exponents Quotient rule for exponents Negative exponents Simplifying Expressions

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 the product of a number a, where 1 ≤ a < 10, and an integer power r of 10: a 10 r Scientific Notation

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Write a Number in Scientific Notation Step 1:Move the decimal point in the original number so that the new number has a value between 1 and 10 Step 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. Step 3:Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. Scientific Notation

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Write each of the following in scientific notation. 4700 1) 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) 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

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. In general, 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

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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

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Martin-Gay, Prealgebra & Introductory Algebra, 3ed 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Operations with Scientific Notation Example 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)

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