Law of Negative Exponents

Law of Negative Exponents

If a power has a negative exponent, move the entire power “to a different floor” and make the exponent positive.
$$2x^{-3}=\frac{2}{x^3}$$

Do not leave the numerator blank; put a 1.$$x^{-3}=\frac{1}{x^3}$$

Powers can move from the denominator to the numerator if they have a negative exponent.$$\frac{2}{5x^{-3}}=\frac{2x^3}{5}$$

Only powers that have a negative exponent should “change floors”: $$\frac{2x^4y^{-2}}{3z^{-1}}=\frac{2x^4z}{3y^2}$$

If the base of the power is a fraction, flip the fraction and make the exponent positive.
$$\Big(\frac{x}{y}\Big)^{-2}=\Big(\frac{y}{x}\Big)^2$$

Common Mistake: A negative exponent DOES NOT change the sign of the base:

$3^{-2} \neq -3^2=-9$

$3^{-2} =\frac{1}{3^2}=\frac{1}{9}$

Practice

EXAMPLE 1

Simplify then evaluate: $2^{11} \div 2^8$

SOLUTION

Using the Law of Division: $$2^{11} \div 2^8=2^{11-8}=2^3$$

Evaluating: $2^3=8$

EXAMPLE 2

Simplify: $(m^0)^{10}$

SOLUTION

Using the Law of Powers: $$(m^0)^{10}=m^0$$

We should more properly use the Law of Zero Exponents to write: $m^0=1$

EXAMPLE 3

Find the value of $x$ that makes the equation true: $4^x \times 4^4=4^7$

SOLUTION

Since we are multiplying the two powers, we use the Law of Multiplication: $$x+4=7$$

Therefore: $x=3$