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:
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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? |
| 4. | 10,000,000,000,000 is 10 raised to what power? |
| 5. | If 14 is equal to 1, then what is 1100? |
