41.7 Signed Bases and Grouping

Signed Bases and Grouping describes how the placement of a negative sign, whether enclosed within parentheses as part of the base or positioned outside the exponent expression entirely, determines whether an exponent applies to a negative value or only to a positive value that is negated afterward, with the parity of the exponent controlling the final sign of the result.

Negative Base inside Parentheses

Description

When a negative sign is enclosed together with a number inside parentheses, and that entire parenthesized quantity carries the exponent, the negative sign is treated as part of the base itself.

Example

In ${(-3)}^2$, the base being raised to the power is $-3$ in its entirety, including the negative sign.

Exterior Negative Sign

Description

When a negative sign appears outside of any parentheses surrounding the base, that sign is treated as a separate multiplication by negative one, applied only after the exponent has already been evaluated on the positive base.

Example

In $-3^2$, only $3$ is raised to the power, and the negative sign is applied to the result afterward.

Even Integer Exponent Sign

Rule

When a negative base, enclosed in parentheses, is raised to an even integer exponent, the result is always positive, since the negative signs pair up completely during the repeated multiplication.

Example

Odd Integer Exponent Sign

Rule

When a negative base, enclosed in parentheses, is raised to an odd integer exponent, the result is always negative, since one negative sign remains unpaired after the repeated multiplication.

Example

Negative Exponent on a Grouped Negative Base

Procedure

When a negative base enclosed in parentheses carries a negative exponent, the reciprocal rule is applied first to the entire grouped base, and the even or odd parity rule then determines the sign of the resulting denominator.

Example

Parenthesized Base Interpretation

Confirming the Grouping

Before evaluating any expression with a negative sign near an exponent, the exact scope of the parentheses is checked first, since this scope determines whether the negative sign belongs to the base or is applied only after the power is evaluated.

Consequence of Misreading Grouping

Misreading ${(-4)}^2$, which equals $16$, as though it were $-4^2$, which equals $-16$, produces an answer with the opposite sign, making correct identification of the grouping an essential first step in evaluating any signed base expression.