7.2 Signed Bases and Power Scope

Signed Bases and Power Scope concerns how the sign of a base behaves under exponentiation, how parentheses determine whether a negative sign is included in the base being raised to a power, and how the parity of the exponent controls the sign of the resulting value.

Powers of Positive Bases

Positive Base Powers

When the base of a power is positive, every product formed by repeated multiplication of that base remains positive regardless of the exponent, since multiplying positive numbers together never introduces a negative sign.

Where the Negative Sign Belongs

Negative Base within Parentheses

When a negative sign is enclosed within parentheses along with the number it applies to, that negative sign is part of the base, and the entire quantity inside the parentheses is what gets raised to the power.

Negative Sign outside a Power

When a negative sign appears outside the parentheses, or immediately before a base with no enclosing parentheses, that sign is not part of the base; the exponent applies only to the positive number, and the negative sign is applied to the result afterward.

Sign of the Result by Exponent Parity

Even Powers of Negative Bases

When a negative base enclosed in parentheses is raised to an even exponent, the result is positive, because the negative signs pair up during repeated multiplication and each pair of negatives produces a positive factor.

Odd Powers of Negative Bases

When a negative base enclosed in parentheses is raised to an odd exponent, the result is negative, because after pairing up as many negative signs as possible, one unpaired negative factor remains and determines the sign of the final product.

Special Base Values

Zero Base with a Positive Exponent

A base of zero raised to any positive exponent equals zero, since repeated multiplication of zero by itself always produces zero regardless of how many factors are used.

Unit Base Powers

A base of one raised to any exponent equals one, since repeated multiplication of one by itself never changes the value away from one.

Negative Unit Base Powers

A base of negative one raised to an exponent alternates between negative one and positive one depending on the parity of the exponent: an even exponent gives positive one, and an odd exponent gives negative one.

Comparing and Confirming Power Scope

Signed Power Notation Comparison

Comparing expressions such as negative three squared with parentheses against the same expression without parentheses highlights how a single pair of parentheses changes the scope of the exponent and therefore the sign and value of the final result, even though the digits and exponent appear identical in both expressions.

Power Scope Verification

Before evaluating a signed power, the scope of the exponent should be verified by checking whether the negative sign lies inside or outside any enclosing parentheses: only a negative sign inside the parentheses that directly surround the base is raised to the power along with that base, while a negative sign outside remains untouched by the exponent and is applied to the evaluated result afterward.