Exponents and powers
Number
Expanded form
Exponential form
Base and exponent
10000
10 × 10 × 10 × 10
104
base 10, exponent 4
base , exponent 5
64
2 × 2 × 2 × 2 × 2 × 2
26
base 2, exponent 6
64
(–2) × (–2) × (–2) × (–2) × (–2) × (–2)
(–2)6
base –2, exponent 6
–32
(–2) × (–2) × (–2) × (–2) × (–2)
(–2)5
base –2, exponent 5
Example 1:
Write the following in exponential form.
a. Minus nine to the power of six
b. One fourth to the power of five
c. Three square to the power of five
Solution:
a. Minus nine to the power of six = (−9)6
b. One fourth to the power of five =
c. Three square to the power of five = (32)5 Example 2:
Write the base and the exponent for the following. a.
b. (–2.5)5 Solution:
a.
Here, base = , exponent = 2 b. (–2.5)5 Here, base = –2.5, exponent = 5 Example 3:
Expand the following expressions.
a. 54
b. (32)3
c. (−2)2 Solution:
a. 54 = 5 × 5 × 5 × 5 b. (32)3 = 32 × 32 × 32 c. (−2)2 = (−2) × (−2) Example 4:
Write the exponents for the base given. a. −125 with (–5) as base
b. 16 with (–2) as base Solution:
a.
−125 = (−5) × (−5) × (−5) = (−5)3
Thus, The exponent is (−5)3. b.
16 = (–2) × (–2) × (–2) × (–2) = (–2)4
Thus, the exponent is (–2)4.
Example 5:
If 400 can be prime factorized as 2x × 52, then what is the value of x?
Solution:
Let us first perform the prime factorization of 400.
2
400
2
200
2
100
2
50
5
25
5
5 1
∴ 400 = 2 × 2 × 2 × 2 × 5 × 5 = 24 × 52
Now, comparing 24 × 52 with 2x × 52, we observe that 24 = 2x
Thus, the value of x is 4.
Example 6:
What is the value of the expression 22 × (−3)3 × (−5)2?
Solution:
22 × (−3)3 × (−5)2 = (2 × 2) × [(−3) × (−3) × (−3)] × [(−5) × (−5)]
= 4 × (−27) × 25
= −2700
Example 7:
How can the number 3960 be written as a product of the powers of its prime factors?
Solution:
3960 = 2 × 2 × 2 × 3 × 3 × 5 × 11 = 23 × 32 × 5 × 11
Example 8:
Express 500 using base 2 and
Expanded form
Exponential form
Base and exponent
10000
10 × 10 × 10 × 10
104
base 10, exponent 4
base , exponent 5
64
2 × 2 × 2 × 2 × 2 × 2
26
base 2, exponent 6
64
(–2) × (–2) × (–2) × (–2) × (–2) × (–2)
(–2)6
base –2, exponent 6
–32
(–2) × (–2) × (–2) × (–2) × (–2)
(–2)5
base –2, exponent 5
Example 1:
Write the following in exponential form.
a. Minus nine to the power of six
b. One fourth to the power of five
c. Three square to the power of five
Solution:
a. Minus nine to the power of six = (−9)6
b. One fourth to the power of five =
c. Three square to the power of five = (32)5 Example 2:
Write the base and the exponent for the following. a.
b. (–2.5)5 Solution:
a.
Here, base = , exponent = 2 b. (–2.5)5 Here, base = –2.5, exponent = 5 Example 3:
Expand the following expressions.
a. 54
b. (32)3
c. (−2)2 Solution:
a. 54 = 5 × 5 × 5 × 5 b. (32)3 = 32 × 32 × 32 c. (−2)2 = (−2) × (−2) Example 4:
Write the exponents for the base given. a. −125 with (–5) as base
b. 16 with (–2) as base Solution:
a.
−125 = (−5) × (−5) × (−5) = (−5)3
Thus, The exponent is (−5)3. b.
16 = (–2) × (–2) × (–2) × (–2) = (–2)4
Thus, the exponent is (–2)4.
Example 5:
If 400 can be prime factorized as 2x × 52, then what is the value of x?
Solution:
Let us first perform the prime factorization of 400.
2
400
2
200
2
100
2
50
5
25
5
5 1
∴ 400 = 2 × 2 × 2 × 2 × 5 × 5 = 24 × 52
Now, comparing 24 × 52 with 2x × 52, we observe that 24 = 2x
Thus, the value of x is 4.
Example 6:
What is the value of the expression 22 × (−3)3 × (−5)2?
Solution:
22 × (−3)3 × (−5)2 = (2 × 2) × [(−3) × (−3) × (−3)] × [(−5) × (−5)]
= 4 × (−27) × 25
= −2700
Example 7:
How can the number 3960 be written as a product of the powers of its prime factors?
Solution:
3960 = 2 × 2 × 2 × 3 × 3 × 5 × 11 = 23 × 32 × 5 × 11
Example 8:
Express 500 using base 2 and