Exploring Powers of 10 Task This activity will help you to learn how to represent very large and very small numbers. Part 1: Discovering the pattern of positive and negative exponents Complete the following table. What pattern do you see? 10 6 = 10 5 = 10 4 = 10 3 = 10 2 = 101 = 10 0 = 10 −1 = 10 −2 =
Part 2: Ways to write large and small numbers Sometimes it can be cumbersome to say and write numbers in their common form (for example trying to display numbers on a calculator with limited screen space). Another option is to use exponential notation and the baseten place value system. Using a calculator (with exponential capabilities), complete each of the following:
1) Explore 10 N for various values of N. Complete the table 105 103 108 109 1010 1012 1015 1018 1020 1025
What patterns do you notice?
2) Enter 45 followed by a string of zeros. How many zeros will your calculator permit? What happens when you press enter? What does 4.5E10 mean? What about 2.3E4? Can you enter this another way?
3) Try sums like (4.5 x 10 N ) + (2 x 10 K ) for different values of N and K. Complete the table (4.5 x 103 ) + (2 x 104 ) (4.5 x 105 ) + (2 x 107 ) (4.5 x 107 ) + (2 x 105 ) (4.5 x 1012 ) + (2 x 109 ) (4.5 x 1015 ) + (2 x 1012 ) (4.5 x 10−2 ) + (2 x 10−3 ) (4.5 x 10−5 ) + (2 x 10−7 ) (4.5 x 10 N ) + (2 x 10 K ) use your own values (4.5 x 10 N ) + (2 x 10 K ) (4.5 x 10 N ) + (2 x 10 K ) Describe the patterns that you notice. 4. What happens if you mix the values of N and K as positives and negatives. Complete the table. (4.5 x 105 ) + (2 x 10−3 ) (4.5 x 108 ) + (2 x 10−5 ) (4.5 x 1012 ) + (2 x 10−12 ) (4.5 x 10−6 ) + (2 x 108 ) (4.5 x 10−12 ) + (2 x 1013 ) Describe the patterns that you notice.
5. Try products like (4.5 x 10 N ) • (2 x 10 K ). Complete the table
(4.5 x 103 ) • (2 x 104 ) (4.5 x 105 ) • (2 x 107 ) (4.5 x 107 ) • (2 x 105 ) (4.5 x 1012 ) • (2 x 109 ). (4.5 x 1015 ) • (2 x 1012 ). Describe any patterns you notice. 6. What happens if you mix values of N and K as positives and negatives? Complete the table (4.5 x 105 ) • (2 x 10−3 ) (4.5 x 108 ) • (2 x 10−5 ) (4.5 x 1012 ) • (2 x 10−12 ) (4.5 x 10−6 ) • (2 x 108 ) (4.5 x 10−12 ) • (2 x 1013 ) Describe any patterns that you notice.
7. Try quotients like (4.5 x 10 N ) ÷ (2 x 10 K ).
Complete the table (4.5 x 103 ) ÷ (2 x 104 ) (4.5 x 105 ) ÷ (2 x 107 ) (4.5 x 107 ) ÷ (2 x 105 ) (4.5 x 1012 ) ÷ (2 x 109 ). (4.5 x 1015 ) ÷ (2 x 1012 ).
Describe any patterns you may notice.
8. What happens as you mix values of N and K as positives and negatives? Complete the table. (4.5 x 105 ) ÷ (2 x 10−3 ) (4.5 x 108 ) ÷ (2 x 10−5 ) (4.5 x 1012 ) ÷ (2 x 10−12 ) (4.5 x 10−6 ) ÷ (2 x 108 ) (4.5 x 10−12 ) ÷ (2 x 1013 ) Describe the patterns you notice.
9. Summarize your findings. Write a paragraph(s) telling about the patterns and how these patterns are alike and different for positive and negative exponents. Tell how they are alike and different if you are adding, multiplying or dividing the exponents? Is a calculator helpful with these types of large or small numbers? Can you work just as efficiently without the calculator once you understand the patterns?