To multiply a number by a power of 10, you shift the decimal point to the right. The number of places you move the decimal is determined by the exponent of 10 or, equivalently, the number of zeros in the power of 10.
Understanding Powers of 10
A power of 10 is simply 10 raised to an exponent (e.g., 102, 103, 106). These exponents indicate how many times to multiply 10 by itself (102 = 10 * 10 = 100). Powers of 10 are also numbers such as 10, 100, 1000, 10000, and so forth.
The Rule: Move the Decimal Point
The core rule is straightforward:
- To multiply by a power of 10, move the decimal point to the right. The reference states: "To multiply by a power of 10, simply move the decimal to the right the same number of places as the exponent or as the number of zeros."
Examples
Let's illustrate this with examples:
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Example 1: 32.5 * 103
- 103 is equal to 1000, which has three zeros.
- Move the decimal point in 32.5 three places to the right: 32.5 → 325 → 3250 → 32500.
- Therefore, 32.5 * 103 = 32,500
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Example 2: 5.12 * 101
- 101 is equal to 10, which has one zero.
- Move the decimal point in 5.12 one place to the right: 5.12 → 51.2
- Therefore, 5.12 * 101 = 51.2
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Example 3: 0.075 * 102
- 102 is equal to 100, which has two zeros.
- Move the decimal point in 0.075 two places to the right: 0.075 → 0.75 → 7.5
- Therefore, 0.075 * 102 = 7.5
Practical Insights
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If you run out of digits to the right of the decimal point, simply add zeros as placeholders.
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Multiplying by powers of 10 makes large numbers easier to express and work with, especially in scientific notation.
Summary Table
| Original Number | Power of 10 | Number of Zeros | Decimal Shift | Result |
|---|---|---|---|---|
| 32.5 | 103 | 3 | 3 places right | 32,500 |
| 5.12 | 101 | 1 | 1 place right | 51.2 |
| 0.075 | 102 | 2 | 2 places right | 7.5 |
