1.05 Powers of 10 with negative exponents
Powers of 10 with negative exponents
Exploration
Some of the powers of 10 we have explored previously are represented in the table.
Try to identify a pattern to complete the rest of the table.
| Power of Ten | Meaning | Value (basic numeral) | In Words |
|---|---|---|---|
| 10^{5} | 10\cdot 10\cdot 10\cdot 10\cdot 10 | 10\,000 | \text{One hundred thousand} |
| 10^{4} | 10\cdot 10\cdot 10\cdot 10 | 10\,000 | \text{Ten thousand} |
| 10^{3} | 10\cdot 10\cdot 10 | 1000 | \text{One thousand} |
| 10^{2} | 10\cdot 100 | 100 | \text{One hundred} |
| 10^{1} | 10 | 10 | \text{Ten} |
| 10^{0} | 1 | 1 | \text{One} |
| 10^{-1} | |||
| 10^{-2} | |||
| 10^{-3} | |||
| 10^{-4} | |||
| 10^{-5} |
- What pattern do you notice in the values of the positive exponents to get from one to the next on the list?
- Apply this pattern to complete the rest of the table. What patterns do you notice in each column for the negative exponents?
From this we can develop the negative exponent property for Base 10, which says 10^{-n} = \dfrac{1}{10^{n}}. In other words, when raising a base to a negative power:
Take the reciprocal of the expression
Turn the power into a positive
Recall for positive powers of 10, we realized that the exponent was the same as the number of zeros after the 1 once evaluated. For negative powers of 10, the exponent relates to the number of zeros between the decimal point and the 1, however not quite in the same way.
What's really happening is the negative exponent is reducing the place value of the number. This makes it look like the decimal point is "moving" to the left a number of places equal to the exponent. But it's really the place value that is changing, not the decimal point.
Examples
Example 1
Rewrite 10^{-7} as both a fraction and a decimal.
The negative exponent property states:10^{-n} = \dfrac{1}{10^{n}}
That is, when raising a base to a negative power:
Take the reciprocal of the expression
Change the sign of the power
This is equivalent to adjusting the place value by n places.