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Exponents and Negative Numbers — Definition, Formula & Examples

Exponents and negative numbers is a topic focused on how the placement of a negative sign relative to parentheses changes the result of exponentiation. The key distinction is whether the negative sign is inside or outside the parentheses being raised to a power.

For any positive real number aa and positive integer nn: (−a)n(-a)^n raises the entire negative value to the nnth power, while −an-a^n is equivalent to −(an)-(a^n), meaning the exponent applies only to aa and the result is then negated. These two expressions are equal when nn is odd but differ when nn is even.

How It Works

Parentheses control what the exponent applies to. In (−3)2(-3)^2, the exponent applies to the entire quantity −3-3, so you multiply (−3)×(−3)=9(-3) \times (-3) = 9. In −32-3^2, the exponent applies only to 33, giving 32=93^2 = 9, and then the negative sign makes it −9-9. This follows the standard order of operations: exponentiation is performed before the negation (which acts like multiplying by −1-1). When the exponent is odd, both forms produce negative results, so the distinction is less dramatic — but when the exponent is even, the results have opposite signs.

Worked Example

Problem: Evaluate both (−4)2(-4)^2 and −42-4^2.
Evaluate (-4)²: The exponent applies to the entire quantity −4. Multiply −4 by itself.
(−4)2=(−4)×(−4)=16(-4)^2 = (-4) \times (-4) = 16
Evaluate -4²: Without parentheses around the negative sign, the exponent applies only to 4. Then negate the result.
−42=−(42)=−(16)=−16-4^2 = -(4^2) = -(16) = -16
Compare: The two expressions differ by sign because of how parentheses direct the exponent.
(−4)2=16but−42=−16(-4)^2 = 16 \quad \text{but} \quad -4^2 = -16
Answer: (−4)2=16(-4)^2 = 16 and −42=−16-4^2 = -16. Parentheses make the difference.

Why It Matters

Misreading the placement of a negative sign with an exponent is one of the most frequent errors on algebra tests and standardized exams. Getting this right is essential when substituting negative values into formulas, such as the quadratic formula or physics equations involving squared terms.

Common Mistakes

Mistake: Treating −32-3^2 as positive 9, assuming the negative sign is included in the base.
Correction: Without parentheses, the exponent binds only to 3. So −32=−(32)=−9-3^2 = -(3^2) = -9. You need (−3)2(-3)^2 to get positive 9.