Section 10.4: Zero and Negative Exponents

Property

$a^0 = 1$, if $a \neq 0$

This definition is based on the second law of exponents. For any non-zero number $a$, the quotient $\frac{a^n}{a^n}$ is equal to 1. Using the law of exponents, we can also write $\frac{a^n}{a^n} = a^{n-n} = a^0$. Therefore, it is logical to define $a^0$ as 1.

Examples

• For a positive integer, $8^0 = 1$.
• For a negative integer, $(-55)^0 = 1$.
• For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$.

Property

A negative exponent indicates the reciprocal of the power with the positive exponent. This means a factor can be moved from the numerator to the denominator (or vice versa) of a fraction by changing the sign of its exponent.

Examples

• To write without a negative exponent: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$.
• For a fraction raised to a negative power, take the reciprocal of the fraction and make the exponent positive: $(\frac{2}{3})^{-4} = (\frac{3}{2})^4 = \frac{81}{16}$.
• To rewrite a fraction using a negative exponent: $\frac{5}{y^2} = 5y^{-2}$.

Explanation

A negative exponent tells you to flip the base to the other side of the fraction bar. An expression like $x^{-n}$ in the numerator becomes $\frac{1}{x^n}$ in the denominator. It's a way to write reciprocals, not to make the number negative.

Property

In decimal place value, positions to the right of the decimal point are represented by negative powers of 10: