Download Identify Bases and Exponents Use the Product and Power Rules for

Math 90
4.1 "Natural-Number Exponents"
Objectives:
*
Identify bases and exponents.
*
Use the product and power rules for exponents.
*
Use the quotient rule for exponents.
Identify Bases and Exponents
Natural-Number Exponents:
A natural-number exponent indicates how many times its base is to be used as a factor.
For any number x and any natural number n;
:
Example 1: (Identifying bases and exponents)
Identify the base and exponent in each expression.
2
a) ( 3)
3
32
b)
c) (5x)
Use the Product and Power Rules for Exponents
Product Rule for Exponents:
For any real number x and any natural numbers m and n;
:
(To multiply exponential expressions with the same base, keep the common base and add the exponents.)
Example 2: (Product rule)
Simplify each expression.
a)
8x4 x3
WARNING!!!
b) a2 b3 a3 b4
Examples of common errors associated with the product rule:
32 34 6= 98
Power Rule for Exponents:
For any real number x and any natural numbers m and n;
23 52 6= 105
:
(To raise an exponential expression to a power, keep the base and multiply the exponents.)
Example 3:
(Power rule)
Simplify each expression.
a)
32
2
b)
x2 x3
6
c)
x2
4
x3
2
Power of a Product and Quotient:
For any real numbers x and y, and any natural number n;
and
where y 6= 0:
(To raise a product to a power, raise each factor of the product to that power. To raise a quotient to a power, raise the
numerator and denominator to that power.)
Page: 1
Notes by Bibiana Lopez
Introductory Algebra by Marvin L. Bittinger
Example 4:
4.1
(Product and Quotient rule)
Simplify each expression.
a)
x2 y
3
b)
x
y2
4
c)
6x3
5y 4
2
Use the Quotient Rule for Exponents
Quotient Rule for Exponents:
To divide exponential expressions with the same base, keep the common base and subtract the exponents.
For any nonzero number x and any integers m and n;
Example 5: (Quotient rule)
Simplify each expression. Write answers using positive exponents.
a5
x4 x3
a) 3
b)
a
x5
c)
x5 y 8
7x2 y 4
Simplify Quotients Raised to Negative Powers
Negative Exponents and Reciprocals:
A fraction raised to a power is equal to the reciprocal of the fraction raised to the opposite power.
For any nonzero real numbers x and y, and any integer n,
Example 6: (Negative exponents and reciprocals)
Simplify each expression. Write answers using positive exponents.
3
y2
a 2 b3
a)
b)
x3
a2 a3 b4
Page: 2
3
Notes by Bibiana Lopez