The numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, ... form a pattern. What is the rule for this pattern?
Answer
This list of numbers results from finding
powers
of 2 in sequence. Look at the table below and you will see several patterns.
Exponential Form
Factor Form
Standard Form
20 =
Any number (except 0) raised to the zero power is always equal to 1.
1
21 =
Any number raised to the first power is always equal to itself.
2
22 =
2 x 2 =
4
23 =
2 x 2 x 2 =
8
24 =
2 x 2 x 2 x 2 =
16
25 =
2 x 2 x 2 x 2 x 2 =
32
26 =
2 x 2 x 2 x 2 x 2 x 2 =
64
27 =
2 x 2 x 2 x 2 x 2 x 2 x 2 =
128
28 =
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =
256
Can you predict the next two numbers in the list after 256?
Answer
Example 1:
Rewrite the numbers 1, 3, 9, 27, 81, 243, ... as a powers of 3.
Solution:
Exponential Form
Standard Form
30 =
1
31 =
3
32 =
9
33 =
27
34 =
81
35 =
243
Can you predict the next two numbers in the list after 243?
Answer
Example 2:
If 73 = 343, then find 74 with only one multiplication.
Solution:
74 = 73 times 7
74 = 343 x 7
74 = 2,401
Example 3:
If 45 = 1,024, then find 46 with only one multiplication.
Solution:
46 = 45 times 4
46 = 1,024 x 4
46 = 4,096
Example 4:
If 100 = 1, and 101 = 10, and 102 = 100, and
103 = 1,000, then predict the values of 106 and 108 in standard form.
Solution:
Exponential Form
Standard Form
Description
100 =
1
The exponent is 0; the number 1 has no zeros.
101 =
10
The exponent is 1; the number 10 has 1 zero.
102 =
100
The exponent is 2; the number 100 has 2 zeros.
103 =
1,000
The exponent is 3; the number 1,000 has 3 zeros.
106 =
1,000,000
The exponent is 6; the number 1,000,000 has 6 zeros.
108 =
100,000,000
The exponent is 8; the number 100,000,000 has 8 zeros.
Summary:
When you find powers of a number in sequence, the resulting list of products forms a pattern.
By examining this pattern, we can predict the next product in the list. Given the standard form
of a number raised to the nth power, we can find the standard form of that number raised to the
n+1 power with a single multiplication. When you find powers of 10 in sequence, a pattern of
zeros is formed in the resulting list of products.
Exercises
Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER.
Do not include commas in your answers.
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
To start over, click CLEAR.
1.
The numbers 1, 5, 25, 125, 625, ... are each powers of what number?
2.
In Exercise 1, what is the next number in the list?
3.
The numbers 1, 6, 36, 216, 1296, ... are each powers of what number?
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