Patterns and Exponents Lesson
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}? 