Transcript Natural-Number Exponents

Polynomials
Copyright © Cengage Learning. All rights reserved.
4
Section
4.1
Natural-Number Exponents
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Write an exponential expression without
exponents.
2 Write a repeated multiplication expression
using exponents.
3 Simplify an expression by using the product
rule for exponents.
4 Simplify an expression by using the power
rules for exponents.
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Objectives
1. Simplify an expression by using the quotient
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2. rules for exponents.
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1. Write an exponential expression
without exponents
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Japanese Sword Making
Japanese sword making
•Japanese sword making
•Begins with raw iron (1:15-3:07)
•Forged and folded 15 times. (4:34-8:06)
How many layers does this produce?
6
Write an exponential expression without exponents
We have used natural-number exponents to indicate
repeated multiplication. For example,
25 = 2  2  2  2  2 = 32
(–7)3 = (–7)(–7)(–7) = –343
x4 = x  x  x  x
–y5 = –y  y  y  y  y
These examples suggest a definition for xn, where n is a
natural number.
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Write an exponential expression without exponents
Natural-Number Exponents
If n is a natural number, then
n factors of x
xn = x  x  x  …  x
In the exponential expression xn, x is called the base and n
is called the exponent. The entire expression is called a
power of x.
Base
xn
Exponent
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Write an exponential expression without exponents
If an exponent is a natural number, it tells how many times
its base is to be used as a factor.
An exponent of 1 indicates that its base is to be used one
time as a factor, an exponent of 2 indicates that its base is
to be used two times as a factor, and so on.
31 = 3
(–4z)2 = (–4z)(–4z)
(–y)1 = –y
(t2)3 = t2  t2  t2
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Example
Find each value to show that
a. (–2)4 and b. –24 have different values.
Solution:
We find each power and show that the results are different.
a. (–2)4 = (–2)(–2)(–2)(–2)
= 16
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Example 1 – Solution
cont’d
b. –24 = –(24)
= –(2  2  2  2)
= –16
Since 16  –16, it follows that –(2)4  –24.
Note that in part a the base is –2 but in part b the base is 2.
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Write an exponential expression without exponents
Comment
There is a pattern regarding even and odd exponents. If the
exponent of a base is even, the result is positive. If the
exponent of a base is odd, the result will be the same sign
as the original base.
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2.
Write a repeated multiplication
expression using exponents
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Example
Write each expression using one exponent.
a. 3  3  3  3  3
b. (5z)(5z)(5z)
Solution:
a. Since 3 is used as a factor five times,
3  3  3  3  3 = 35
b. Since 5z is used as a factor three times,
(5z)(5z)(5z) = (5z)3
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3.
Simplify an expression by using
the product rule for exponents
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Simplify an expression by using the product rule for exponents
To develop a pattern for multiplying exponential
expressions with the same base, we consider the product
x2  x3.
Since the expression x2 means that x is to be used as a
factor two times and the expression x3 means that x is to be
used as a factor three times, we have
2 factors of x
x2x3 = x  x

3 factors of x
xxx
5 factors of x
=xxxxx
= x5
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Simplify an expression by using the product rule for exponents
In general,
m factors of x
n factors of x
xm  xn = x  x  x  . . .  x  x  x  x  . . .  x
m + n factors of x
= x  x  x  x  x  x  ...  x  x  x
= xm + n
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Simplify an expression by using the product rule for exponents
This discussion suggests the following pattern: To multiply
two exponential expressions with the same base, keep the
base and add the exponents.
Product Rule for Exponents
If m and n are natural numbers, then
xmxn = xm + n
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Example
Simplify each expression.
a. x3x4 = x3 + 4
Keep the base and add the exponents.
= x7
b. y2y4y = (y2y4)y
3+4=7
Use the associative property to group
y2 and y4 together.
= (y2 + 4)y
Keep the base and add the exponents.
= y6y
2+4=6
= y6 + 1
Keep the base and add the exponents:
y = y1.
= y7
6+1=7
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Simplify an expression by using the product rule for exponents
Comment
The product rule for exponents applies only to exponential
expressions with the same base. An expression such as
x2y3 cannot be simplified, because x2 and y3 have different
bases.
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4.
Simplify an expression by using
the power rules for exponents
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Simplify an expression by using the power rules for exponents
To find another pattern of exponents, we consider the
expression (x3)4, which can be written as x3  x3  x3  x3.
Because each of the four factors of x3 contains three
factors of x, there are 4  3 (or 12) factors of x. Thus, the
expression can be written as x12.
(x3)4 = x3  x3  x3  x3
12 factors of x
=xxxxxxxxxxxx
x3
x3
x3
x3
= x12
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Simplify an expression by using the power rules for exponents
In general,
n factors of xm
(xm)n = xm  xm  xm  . . .  xm
m  n factors of x
= x  x  x  x  x  x  x  ...  x
= xm  n
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Simplify an expression by using the power rules for exponents
The previous discussion suggests the following pattern: To
raise an exponential expression to a power, keep the base
and multiply the exponents.
Power Rule for Exponents
If m and n are natural numbers, then
(xm)n = xmn
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Example
Simplify:
a. (23)7 = 23  7
= 221
b. (z7)7 = z7  7
= z49
Keep the base and multiply the exponents.
3  7 = 21
Keep the base and multiply the exponents.
7  7 = 49
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Simplify an expression by using the power rules for exponents
To find more patterns for exponents, we consider the
expressions (2x)3 and
.
(2x)3 = (2x)(2x)(2x)
= (2  2  2)(x  x  x)
= 23x3
= 8x3
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Simplify an expression by using the power rules for exponents
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Simplify an expression by using the power rules for exponents
These examples suggest the following patterns: To raise a
product to a power, we raise each factor of the product to
that power, and to raise a quotient to a power, we raise
both the numerator and denominator to that power.
Product to a Power Rule for Exponents
If n is a natural number, then
(xy)n = xnyn
Quotient to a Power Rule for Exponents
If n is a natural number, and if y  0, then
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5.
Simplify an expression by using
the quotient rule for exponents
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Simplify an expression by using the quotient rule for exponents
To find a pattern for dividing exponential expressions, we
consider the fraction
where the exponent in the
numerator is greater than the exponent in the denominator.
We can simplify the fraction as follows:
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Simplify an expression by using the quotient rule for exponents
The result of 43 has a base of 4 and an exponent of
5 – 2 (or 3). This suggests that to divide exponential
expressions with the same base, we keep the base and
subtract the exponents.
Quotient Rule for Exponents
If m and n are natural numbers, m  n and x  0, then
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Example
Simplify. Assume no division by zero.
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Example
cont’d
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