264 CHAPTER 5 Exponents, Polynomials, and Polynomial Functions
OBJECTIVES Recall that we use exponents to write products of repeated factors. For example,
is defined as
The number 5, the exponent,shows that the base2 appears as a factor five times.
The quantity is called an exponentialor a power.We read as “2 to the fifth
power”or “2 to the fifth.”
25 25
25 2 # 2 # 2 # 2 # 2 =32.
Integer Exponents and Scientific Notation
5.1
1 Use the product
rule for exponents.
2 Define 0 and
negative exponents.
3 Use the quotient
rule for exponents.
4 Use the power rules
for exponents.
5 Simplify exponential
expressions.
6 Use the rules for
exponents with
scientific notation.NOW TRY
EXERCISE 1
Apply the product rule, if
possible, in each case.
(a)
(b)
(c)p^2 #q^2
15 x^4 y^721 - 7 xy^3285 # 84
OBJECTIVE 1 Use the product rule for exponents.Consider the product
, which can be simplified as follows.
This result, that products of exponential expressions with the same baseare found by
adding exponents, is generalized as the product rule for exponents.
25 # 23 = 12 # 2 # 2 # 2 # 2212 # 2 # 22 = 28
5 + 3 = 825 # 23
Product Rule for ExponentsIf mand nare natural numbers and ais any real number, then
That is, when multiplying powers of like bases, keep the same base and add the
exponents.
a m#a na mn.
To see that the product rule is true, use the definition of an exponent.
aappears as a factor mtimes. aappears as a factor ntimes.mfactors nfactorsfactors
am#an=am+n1 m+n 2=a#a#a#...#a
am#an=a#a#a#...#a#a#a#a#...#a
am =a#a#a#...#a an=a#a#a#...#a
⎧⎪⎪⎪⎨⎪⎪⎪⎩⎧⎪⎪⎪⎨⎪⎪⎪⎩⎧⎪⎪⎪⎨⎪⎪⎪⎩⎧⎪⎪⎪⎨⎪⎪⎪⎩⎧⎪⎪⎪⎨⎪⎪⎪⎩Using the Product Rule for ExponentsApply the product rule for exponents, if possible, in each case.
(a)
(b) (c)
(d)
Commutative property
Multiply; product rule=- 15 y^6
=- 15 y^2 +^4
= 51 - 32 y^2 y^4
15 y^221 - 3 y^42
53 # 5 = 53 # 51 = 53 +^1 = 54 y^3 #y^8 #y^2 = y^3 +^8 +^2 =y^13
34 # 37 = 34 +^7 = 311
EXAMPLE 1
Do notmultiply the bases.
Keep the same base.(e)
= 14 p^8 q^3
= 14 p^3 +^5 q^1 +^2
= 7122 p^3 p^5 q^1 q^2
17 p^3 q 212 p^5 q^22
(f )x^2 #y^4 The product rule does not apply because the bases are not the same.
NOW TRY ANSWERS
- (a) (b)
(c)The product rule does not
apply.
89 - 35 x^5 y^10NOW TRY