# Investigation: Patterns in powers of 10

Lesson

### Objective

Identify patterns in powers of 10 that can be used to calculate more quickly

### Vocabulary

When you are discussing the patterns that you are noticing, be sure to use the vocabulary below in context:

Remember!

#### Three different forms of the same number

• This may be called exponential form $10^3$103
• This may be called expanded form $10\times10\times10$10×10×10
• This may be called standard form $1000$1000

### Procedure

1. Complete row 1, row 2, and row 3 in the table below.  Are there any patterns that you are noticing? What is happening to the standard form when the exponent of the number written in exponential form increases? What is happening to the standard form when the exponent of the number written in exponential form decreases?
2. Turn and talk to a partner about what patterns you noticed while completing the table for row 1, row 2, and row 3.
3. Use the pattern that you established when considering row 1, row 2 and row 3 to determine the standard form of the number in row 4.  Now use the scientific calculator to verify that your answer is correct.
4. Write a "rule" that you think might work to determine the value of any number raised to the zero power and test this rule be raising bases other than 10 to the zero power.
row EXPONENTIAL FORM EXPANDED FORM STANDARD FORM
(1) $10^3$103
(2) $10^2$102
(3) $10^1$101
(4) $10^0$100

5. Next, use the same pattern you used to establish standard form in row 4, above, to determine standard form in row 5, row 6 and row 7 of the table below.  Use a calculator to verify that your responses are correct.  (Be sure to write the numbers both in fraction and decimal form when you are completing row 5, row 6 and row 7).

6. Is there a general rule that you could use to describe the result of raising 10 to a negative exponent?

row EXPONENTIAL FORM STANDARD FORM (FRACTION) STANDARD FORM (DECIMAL)
(5) $10^{-1}$101
(6) $10^{-2}$102
(7) $10^{-3}$103
(8) $10^{-n}$10n

### Discussion questions

1. Can you write a general rule that you can use to calculate the value of $10^n$10n, for example $10^7$107?
2. Can you write a general rule that you can use to calculate the value of $a^0$a0, for example $10^0$100?
3. Can you write a general rule that you can use to calculate the value of $10^{-n}$10n, for example $10^{-5}$105?
4. Why do you think we use exponent notation if it is just repeated multiplication?