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

Wrong Answer

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

Right Answer

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

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