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**Copyright © 2013, 2009, 2006 Pearson Education, Inc.**

Section 5.7 Negative Exponents and Scientific Notation Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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**Negative Exponents in Numerators and Denominators**

If b is any real number other than 0 and n is a natural number, then When a negative number appears as an exponent, switch the position of the base (from numerator to denominator or denominator to numerator) and make the exponent positive. The sign of the base does not change.

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Negative Exponents EXAMPLES

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Negative Exponents

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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Objective #1: Example

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**Simplifying Exponential Expressions**

An exponential expression is “simplified” when a. Each base occurs only once. b. No parentheses appear. c. No powers are raised to powers. d. No negative or zero exponents appear.

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**Simplification Techniques**

Simplifying Exponential Expressions Simplification Techniques Examples If necessary, remove parentheses by using the Products to Powers Rule or the Quotient to Powers Rule. If necessary, simplify powers to powers by using the Power Rule.

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**Simplification Techniques**

Simplifying Exponential Expressions Simplification Techniques Examples Be sure each base appears only once in the final form by using the Product Rule or Quotient Rule If necessary, rewrite exponential expressions with zero powers as 1. Furthermore, write the answer with positive exponents by using the Negative Exponent Rule

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**Simplifying Exponential Expressions**

EXAMPLE Cube each factor in the numerator. Multiply powers using (bm)n = bmn. Division with the same base, subtract exponents. When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.

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**Simplifying Exponential Expressions**

EXAMPLE Cube each factor in the numerator. Multiply powers using (bm)n = bmn. Division with the same base, subtract exponents. When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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Objective #2: Example

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**Scientific Notation Scientific Notation**

At times you may find it necessary to work with really large numbers, or alternately, really small numbers. Here you will learn how to write these often cumbersome numbers in scientific notation. Scientific Notation A positive number is written in scientific notation when it is expressed in the form Where a is a number greater than or equal to 1 and less than 10 and n is an integer.

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**Converting from Scientific to Decimal Notation**

Scientific Notation Converting from Scientific to Decimal Notation We can use n, the exponent on the 10 in , to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left places.

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Objective #3: Example

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Objective #3: Example

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**Converting from Decimal to Scientific Notation**

Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10. Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal was moved. The exponent n is positive if the given number is great than 10 and negative if the given number is between 0 and 1.

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Objective #4: Example

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Objective #4: Example

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**Computations with Scientific Notation**

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**Computations with Scientific Notation**

EXAMPLE Perform the indicated computation, writing the answer in scientific notation. SOLUTION Regroup factors Multiply Simplify Rewrite in scientific notation

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**Computations with Scientific Notation**

EXAMPLE Perform the indicated computation, writing the answer in scientific notation. SOLUTION Group factors Divide Simplify Write in scientific notation

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Objective #5: Example

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Objective #5: Example

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Objective #5: Example

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Objective #5: Example

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Objective #5: Example

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Objective #5: Example

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Scientific Notation EXAMPLE The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia? SOLUTION We will first write 7.2 million in scientific notation. It is: Now we can answer the question.

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Scientific Notation EXAMPLE The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia? SOLUTION We will first write 7.2 million in scientific notation. It is: Now we can answer the question.

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**Scientific Notation Divide ‘dollar amount’ by ‘acreage’ Simplify**

CONTINUED Divide ‘dollar amount’ by ‘acreage’ Simplify Divide Subtract Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.

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**Scientific Notation Divide ‘dollar amount’ by ‘acreage’ Simplify**

CONTINUED Divide ‘dollar amount’ by ‘acreage’ Simplify Divide Subtract Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.

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Objective #6: Example

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Objective #6: Example

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