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
Worked examples
Evaluate 5^-3
Rewrite the power as a reciprocal.
5^-3 = 1/5^35^3 = 125
Result: 5^-3 = 1/125 = 0.008
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
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 != 0Negative 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^3Where this idea is used
- Writing small quantities such as 10^-6 meters.
- Simplifying rates and inverse-power relationships.
- Converting scientific notation with negative powers of ten.
Common mistakes
- Changing 2^-3 into -2^3 instead of 1/2^3.
- Allowing a zero base even though the reciprocal would divide by zero.
- Moving a sum across a fraction bar as if it were one multiplicative factor.
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.
Can zero have a negative exponent?
No. It would require division by zero.
Does the negative sign belong to the base?
Only when parentheses include it, as in (-2)^4.