Algebra
Negative Exponents Explained
A negative exponent does not make a value negative. It moves a nonzero base to the opposite side of a fraction: a^(-n) = 1/a^n.
Reviewed
Follow the pattern through exponent zero
Moving one place to the right divides the value by the base. Continuing the same pattern past exponent zero forces negative powers to be reciprocals.
| Exponent | Power of 2 | Value |
|---|---|---|
| 3 | 2^3 | 8 |
| 2 | 2^2 | 4 |
| 1 | 2^1 | 2 |
| 0 | 2^0 | 1 |
| -1 | 2^-1 | 1/2 |
| -2 | 2^-2 | 1/4 |
| -3 | 2^-3 | 1/8 |
The rule a^-n = 1/a^n requires a not equal to zero; 0 raised to a negative exponent would require division by zero.
Why a negative exponent means reciprocal
The quotient rule says a^m/a^n = a^(m-n). When n is greater than m, canceling factors leaves powers of a in the denominator. The negative exponent is compact notation for that reciprocal.
For example, a^2/a^5 cancels two factors and leaves 1/a^3. The exponent calculation gives a^(2-5) = a^-3, so the two expressions must agree.
a^(-n) = 1/a^n, for a != 0Worked example 1
Evaluate 5^-3
Rewrite the power as a reciprocal.
5^-3 = 1/5^35^3 = 125
Result: 5^-3 = 1/125 = 0.008
Negative bases and negative exponents
The exponent sign and the base sign do different jobs. (-2)^-3 is the reciprocal of (-2)^3, so it equals -1/8. By contrast, -2^-3 is parsed as -(2^-3) and also equals -1/8; an even exponent would reveal the difference.
Use parentheses whenever the sign is intended to be part of the base.
Moving factors across a fraction bar
A factor with a negative exponent can be rewritten with a positive exponent on the other side of the fraction bar. Move the entire factor, not just the exponent.
Restrictions remain important. If a variable appears in a denominator at any stage, its value cannot be zero.
x^-3 / y^-2 = y^2 / x^3Worked example 2
Simplify 6x^-2 y^3
Move only the x factor to the denominator.
x^-2 = 1/x^2Keep 6 and y^3 in the numerator
Result: 6y^3/x^2, with x != 0
Questions about negative exponents
Is a negative exponent always a small number?
Not always. If the base has magnitude below 1, taking its reciprocal can produce a large value.