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 
2^{0} =  Any number (except 0) raised to the zero power is always equal to 1.  1 
2^{1} =  Any number raised to the first power is always equal to itself.  2 
2^{2} =  2 x 2 =  4 
2^{3} =  2 x 2 x 2 =  8 
2^{4} =  2 x 2 x 2 x 2 =  16 
2^{5} =  2 x 2 x 2 x 2 x 2 =  32 
2^{6} =  2 x 2 x 2 x 2 x 2 x 2 =  64 
2^{7} =  2 x 2 x 2 x 2 x 2 x 2 x 2 =  128 
2^{8} =  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 
3^{0} =  1 
3^{1} =  3 
3^{2} =  9 
3^{3} =  27 
3^{4} =  81 
3^{5} =  243 
Can you predict the next two numbers in the list after 243? Answer
Example 2: If 7^{3} = 343, then find 7^{4} with only one multiplication.
Solution: 7^{4} = 7^{3} times 7
7^{4} = 343 x 7
7^{4} = 2,401
Example 3: If 4^{5} = 1,024, then find 4^{6} with only one multiplication.
Solution: 4^{6} = 4^{5} times 4
4^{6} = 1,024 x 4
4^{6} = 4,096
Example 4: If 10^{0} = 1, and 10^{1} = 10, and 10^{2} = 100, and 10^{3} = 1,000, then predict the values of 10^{6} and 10^{8} in standard form.
Solution:

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 n^{th} 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 1^{4} is equal to 1, then what is 1^{100}? 