1.7 Prime Factors and Exponents 855 Use exponential notation.
In Example 6, we saw that the prime factorization of 280 is Because
this factorization has three factors of 2, we call 2 a repeated factor.We can use
exponential notationto write 2 2 2 in a more compact form.
2 2 2 5 7.
Exponent and BaseAn exponentis used to indicate repeated multiplication. It tells how many
times the baseis used as a factor.The exponent is 3.Read as “2 to the third power” or “2 cubed.”Repeated factors The base is 2.The prime factorization of 280 can be written using exponents:
In the exponential expression the number 2 is the base and 3 is the exponent.
The expression itself is called a power of 2.
23 ,
23 5 7.
2 2 2 5 7
⎫⎪⎬⎪⎭
2 2 2 23 23EXAMPLE (^7) Write each product using exponents:
a. b. c.
StrategyWe will determine the number of repeated factors in each expression.
WHYAn exponent can be used to represent repeated multiplication.
Solution
a.The factor 5 is repeated 4 times. We can represent this repeated multiplication
with an exponential expression having a base of 5 and an exponent of 4:
b. 7 is used as a factor 2 times.
c. 2 is used as a factor 4 times, and 3 is
used as a factor 3 times.
2(2)(2)(2)(3)(3)(3) 24 (3^3 )
7 7 11 72 11
5 5 5 5 54
5 5 5 5 7 7 11 2(2)(2)(2)(3)(3)(3)
Self Check 7
Write each product using
exponents:
a.
b.
c.
Now TryProblems 77 and 812 2 2 3 3 5
5(5)(7)(7)
3 3 7
6 Evaluate exponential expressions.
We can use the definition of exponent to evaluate(find the value of) exponential
expressions.
EXAMPLE (^8) Evaluate each expression:
a. b. c. d.
StrategyWe will rewrite each exponential expression as a product of repeated
factors, and then perform the multiplication. This requires that we identify the
base and the exponent.
WHYThe exponent tells the number of times the base is to be written as a
factor.
Solution
We can write the steps of the solutions in horizontal form.
72 25 104 61
Self Check 8
Evaluate each expression:
a. b.
c. d.
Now TryProblem 89