Product Rule for Exponents (The Basic Rule of Exponents)
The powers of 10 stand in a multiplicative relationship to each other. If we multiply one of them by 10, we get the next one:
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101 x 10 = 10 x 10 = 102 |
102 x 10 = (10 x 10) x 10 = 103, |
In general, 10m x 10 = 10m+1, will also be valid and demonstrated with any non-zero integer m. In fact, any product of powers of 10 is another power of 10. For example:
Example 1 (Product Rule with Positive Exponents)
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Solution I |
Solution II |
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103 × 105 = 10(3 + 5) = 108 |
103 × 105 = (10 × 10 × 10) × (10 × 10 × 10 × 10 × 10) = 108 |
Solution I & II Rationale:
In both these examples, solution I is the formal application of the Product Rule, and solution II is the justification in terms of the definition of powers.
Common Misconceptions about the Product Rule: When traditionally presented as problems students will be tempted to multiply the base as well as the exponent. However, if they are asked to compute as in Solution II, they may see that the base remains the same, only the number of times it is used as a factor (i.e., the exponent) changes.
As this example shows, there is a very simple and pretty relationship between the exponents of the two factors and the exponent of the product. The total number of 10s in the product is just the sum of the number of 10s in the two factors. This is summarized by the Law of Exponents, also known as the
Product rule for exponents: 10m x 10n = 10m+n, for any whole numbers m and n.
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