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Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

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1.4 Exponential and Scientific Notation ■ The Product and Quotient Rules ■ The Zero Exponent ■ Negative Integers as Exponents ■ Simplifying (a m ) n ■ Raising a Product or a Quotient to a Power ■ Scientific Notation ■ Significant Digits and Rounding

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Slide 1- 3 Copyright © 2012 Pearson Education, Inc. Multiplying with Like Bases: The Product Rule For any number a and any positive integers m and n, (When multiplying powers, if the bases are the same, keep the base and add the exponents.)

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Slide 1- 4 Copyright © 2012 Pearson Education, Inc. Example Solution Multiply and simplify:

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Slide 1- 5 Copyright © 2012 Pearson Education, Inc. Dividing with Like Bases: The Quotient Rule For any nonzero number a and any positive integers m and n, m > n, (When dividing powers, if the bases are the same, keep the base and subtract the exponents of the denominator from the exponent of the numerator.)

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Slide 1- 6 Copyright © 2012 Pearson Education, Inc. Example Solution Divide and simplify:

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Slide 1- 7 Copyright © 2012 Pearson Education, Inc. The Zero Exponent For any nonzero real number a, (Any nonzero number raised to the zero power is 1. 0 0 is undefined.)

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Slide 1- 8 Copyright © 2012 Pearson Education, Inc. Example Solution Evaluate each of the following for y = 5:

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Slide 1- 9 Copyright © 2012 Pearson Education, Inc. Integer Exponents For any real number a that is nonzero and any integer n, (The numbers a -n and a n are reciprocals of each other.)

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Slide 1- 10 Copyright © 2012 Pearson Education, Inc. Example Express using positive exponents and simplify if possible. Solution

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Slide 1- 11 Copyright © 2012 Pearson Education, Inc. Factors and Negative Exponents For any nonzero real numbers a and b and any integers m and n, (A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.)

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Slide 1- 12 Copyright © 2012 Pearson Education, Inc. Example Write an equivalent expression without negative exponents: Solution

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Slide 1- 13 Copyright © 2012 Pearson Education, Inc. Example The product and quotient rules apply for all integer exponents. Solution Simplify:

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Slide 1- 14 Copyright © 2012 Pearson Education, Inc. The Power Rule For any real number a and any integers m and n, (To raise a power to a power, multiply the exponents.)

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Slide 1- 15 Copyright © 2012 Pearson Education, Inc. Example Simplify: Solution

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Slide 1- 16 Copyright © 2012 Pearson Education, Inc. Raising a Product to a Power For any integer n, and any real numbers a and b for which (ab) n exists, (To raise a product to a power, raise each factor to that power.) Raising a Product or a Quotient to a Power

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Slide 1- 17 Copyright © 2012 Pearson Education, Inc. Example Simplify: Solution

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Slide 1- 18 Copyright © 2012 Pearson Education, Inc. Raising a Quotient to a Power For any integer n, and any real numbers a and b for which a/b, a n, and b n exist, (To raise a quotient to a power, raise both the numerator and denominator to that power.)

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Slide 1- 19 Copyright © 2012 Pearson Education, Inc. Example Simplify: Solution

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Slide 1- 20 Copyright © 2012 Pearson Education, Inc. Definitions and Properties of Exponents The following summary assumes that no denominators are 0 and that 0 0 is not considered. For any integers m and n, 1 as an exponent:a 1 = a 0 as an exponent:a 0 = 1 Negative Exponents: The Product Rule: The Quotient Rule: The Power Rule:(a m ) n = a mn Raising a product to a power:(ab) n = a n b n Raising a quotient to a power:

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Slide 1- 21 Copyright © 2012 Pearson Education, Inc. Scientific Notation Scientific notation for a number is an expression of the form, where N is in decimal notation, and m is an integer. Scientific Notation

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Slide 1- 22 Copyright © 2012 Pearson Education, Inc. Converting Decimal Notation Scientific Notation

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Slide 1- 23 Copyright © 2012 Pearson Education, Inc. Convert to decimal notation: a) 3.842 10 6 b) 5.3 10 7 Solution a) Since the exponent is positive, the decimal point moves right 6 places. 3.842000. 3.842 10 6 = 3,842,000 b) Since the exponent is negative, the decimal point moves left 7 places. 0.0000005.3 5.3 10 7 = 0.00000053 Example

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Slide 1- 24 Copyright © 2012 Pearson Education, Inc. Write in scientific notation: a) 94,000b) 0.0423 Solution a) We need to find m such that 94,000 = 9.4 10 m. This requires moving the decimal point 4 places to the right. 94,000 = 9.4 10 4 b) To change 4.23 to 0.0423 we move the decimal point 2 places to the left. 0.0423 = 4.23 10 2 Example

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Slide 1- 25 Copyright © 2012 Pearson Education, Inc. Significant Digits and Rounding When two or more measurements written in scientific notation are multiplied or divided, the result should be rounded so that it has the same number of significant digits as the measurement with the fewest significant digits. Rounding should be performed at the end of the calculation.

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Slide 1- 26 Copyright © 2012 Pearson Education, Inc. Significant Digits and Rounding When two or more measurements written in scientific notation are added or subtracted, the result should be rounded so that it has as many decimal places as the measurement with the fewest decimal places.

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Slide 1- 27 Copyright © 2012 Pearson Education, Inc. Example Multiply and write scientific notation for the answer: Solution rounded to 2 significant digits:

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Slide 1- 28 Copyright © 2012 Pearson Education, Inc. Divide and write scientific notation for the answer:. Solution Example

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Slide 1- 29 Copyright © 2012 Pearson Education, Inc. Use a graphing calculator to calculate (–5.2) 3. Solution Example

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Slide 1- 30 Copyright © 2012 Pearson Education, Inc. Use a graphing calculator to calculate (6.2 × 10 3 ) (3.1 × 10 –12 ). Solution Example

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