copyright © 2011 pearson education, inc. publishing as prentice hall. 3.2 negative exponents and...
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Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
3.2
Negative Exponents and Scientific Notation
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,
0
2646
4
x
xxx
x
But what does x -2 mean?
26
4 11
xxxxxxxxx
xxxx
x
x
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
nn
aa
1
Negative Exponents
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
23
1
9
1
7
1
x
4
2
x
23
7x
42 x
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.
Don’t forget that since there are no parentheses, x is the base for the exponent –4.
Helpful HintHelpful Hint
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
2323
1
9
1
2)3( 2)3(
1
9
1
31x
3x
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.
3
1x
1)
3
11
x
2) 4
2
y
x
4
2
1
1
y
x2
4
x
y
1
3x 3x
(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
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 am · an = am+n
Power Rule for exponents (am)n = amn
Power of a Product (ab)n = an · bn
Power of a Quotient 0,
bb
a
b
an
nn
Quotient Rule for exponents 0, aaa
a nmn
m
Zero exponent a0 = 1, a ≠ 0
Negative exponent 0,1
aa
an
n
Summary of Exponent Rules
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.
3 2 a3b
3 4 a7b 3
2
3 2 a3b 2
3 4 a7b 3 2
Power of a quotient rule
3 2 2
a3 2b 2
3 4 2a7 2
b 3 2
Power of a product rule
34 a 6b2
38 a 14b6
Power rule for exponents
34 8 a14 6b2 6
Quotient rule for exponents
3 4 a8b 4 a8
81b434 a14b2
38 a6b6
Negative exponents
a8
34 b4
Negative exponents
Simplifying Expressions
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 10r
Scientific Notation
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1010
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
To Write a Number in Scientific NotationStep 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
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1111
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Write each of the following in scientific notation.
47001) 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 = 4.7 103
0.000472) 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
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1212
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
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1313
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Write each of the following in standard notation. 5.2738 1031)
Since the exponent is a positive 3, we move the decimal 3 places to the right.
5.2738 103 = 5273.8
6.45 10-52)
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
Martin-Gay, Prealgebra & Introductory Algebra, 3ed 1414
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 · 105) = 59.13 103
= 59,130
(7.3 10-2)(8.1 105)1)
2) 9
4
104
102.1
9
4
10
10
4
2.1 5103.0 000003.0