Negative exponents are a compact way to show reciprocals, which are numbers that flip across a fraction bar. They matter because powers with negative exponents appear in algebra, scientific notation, formulas, and graphs. Learning the rule helps you simplify expressions without treating the negative sign as part of the base.
The key idea is that a negative exponent tells you where the factor belongs in a fraction, not that the value must be negative.
The main rule is a^-n = 1/a^n, where a is not 0. If a power is in the numerator with a negative exponent, move the base and exponent to the denominator and make the exponent positive. If a power is in the denominator with a negative exponent, move it to the numerator and make the exponent positive.
Zero exponents fit the same pattern because any nonzero base to the zero power equals 1.
Key Facts
- Negative exponent rule: a^-n = 1/a^n, where a ≠ 0.
- Denominator rule: 1/a^-n = a^n, where a ≠ 0.
- Zero exponent rule: a^0 = 1, where a ≠ 0.
- Product rule: a^m · a^n = a^(m+n), using the same nonzero base.
- Quotient rule: a^m/a^n = a^(m-n), where a ≠ 0.
- Power of a power rule: (a^m)^n = a^(mn).
Vocabulary
- Negative exponent
- A negative exponent tells you to write the reciprocal of the base raised to the matching positive exponent.
- Reciprocal
- The reciprocal of a nonzero number is 1 divided by that number, such as the reciprocal of 5 being 1/5.
- Base
- The base is the number or expression being raised to a power.
- Exponent
- The exponent tells how many times the base is used as a factor, or how that power should be simplified.
- Zero exponent
- A zero exponent means the value is 1 as long as the base is not 0.
Common Mistakes to Avoid
- Changing 3^-2 into -9 is wrong because the negative sign in the exponent does not make the value negative. The correct simplification is 3^-2 = 1/3^2 = 1/9.
- Forgetting the condition a ≠ 0 is wrong because division by zero is undefined. Expressions like 0^-2 and 1/0^2 are not allowed.
- Moving only the exponent across the fraction bar is wrong because the base with its exponent must move together. For example, x^-3 becomes 1/x^3, not x/3.
- Treating a^0 as 0 is wrong because any nonzero base to the zero power equals 1. For example, 7^0 = 1, not 0.
Practice Questions
- 1 Simplify 4^-3 as a fraction.
- 2 Simplify (2x^-3y^0)/(5x^-1) using positive exponents only.
- 3 Explain why 6^-2 is positive even though it has a negative exponent.