Definition: The laws of exponents are rules for simplifying powers, such as adding exponents when multiplying powers with the same base.
What are the laws?
For a nonzero base a:
- Product: am x an = am+n
- Quotient: am / an = am-n
- Power of a power: (am)n = amn
- Zero exponent: a0 = 1
- Negative exponent: a-n = 1 / an
Why do they work?
Write it out: 23 x 24 = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 27.
What does an example look like?
Simplify (35 x 3-2) / 34. The numerator is 33. Then 33 / 34 = 3-1 = 1/3.
What about a negative exponent?
5-2 = 1 / 25. It does not make the answer negative.
What is a common mistake?
Multiplying bases. In 42 x 43, the base stays 4, so the answer is 45 = 1,024, not 165. Also, the product rule needs equal bases: 23 x 33 does not combine that way.
Example: (76) / (74) = 72 = 49.