Lesson 4: Negative Exponents

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

If $a$ and $b$ are real numbers, and $m$ and $n$ are integers, then:
Product Property: $a^m \cdot a^n = a^{m+n}$
Power Property: $(a^m)^n = a^{mn}$
Product to a Power: $(ab)^m = a^m b^m$
Quotient Property: $\frac{a^m}{a^n} = a^{m-n}, a \neq 0$
Zero Exponent Property: $a^0 = 1, a \neq 0$
Quotient to a Power Property: $(\frac{a}{b})^m = \frac{a^m}{b^m}, b \neq 0$
Properties of Negative Exponents: $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$
Quotient to a Negative Exponent: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$

Examples

• Using the Product Property: $x^5 \cdot x^{-2} = x^{5+(-2)} = x^3$.

• Using the Power Property: $(y^{-3})^4 = y^{-3 \cdot 4} = y^{-12} = \frac{1}{y^{12}}$.