1Cademy - Simplifying $$(-2)^4$$ and $$-2^4$$

Learn Before

$(-a)^n$ Versus $-a^n$

Example

Simplifying $(-2)^4$ and $-2^4$

These two expressions illustrate how parentheses around a negative base change the result of exponentiation:

$(-2)^4 = 16$ : The base is $-2$ . Expand as four factors of $-2$ : $(-2)(-2)(-2)(-2)$ Multiply step by step: $(-2)(-2) = 4$ , then $4(-2) = -8$ , then $(-8)(-2) = 16$ . An even number of negative factors produces a positive result.

$-2^4 = -16$ : Only $2$ is the base; the expression means "the opposite of $2^4$ ." Expand $2^4$ first: $-(2 \cdot 2 \cdot 2 \cdot 2)$ Multiply step by step: $2 \cdot 2 = 4$ , then $4 \cdot 2 = 8$ , then $8 \cdot 2 = 16$ . Apply the negation: $-16$ .

Although the two expressions differ by only a pair of parentheses, $(-2)^4 = 16$ is positive while $-2^4 = -16$ is negative — a difference that hinges entirely on whether the exponent acts on the negative number or only on the positive number.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After