ACT Math: Negative Exponents, Minus Signs, and Parentheses

A negative exponent tells you to take a reciprocal. Parentheses tell you whether a minus sign belongs to the base being raised to a power. A minus sign outside that power is applied afterward. Identify the base before calculating, and keep those jobs separate.

For example, $(-6)^2=36$, $-6^2=-36$, and $6^{-2}=\frac{1}{36}$. The position of the minus sign changes the operation.

ACT’s official mathematics content guide includes integer and rational exponents in Number and Quantity. This article focuses on integer exponents. Every practice problem below is original Kibo material, not an official ACT item.

Read the base before doing the arithmetic

In $(-6)^2$, the parentheses make the entire negative number the base. Squaring it multiplies two negative factors. In $-6^2$, the base of the power is $6$; the leading minus remains outside.

$$(-6)^2=(-6)(-6)=36$$

$$-6^2=-(6\times6)=-36$$

This follows the order of operations explained by OpenStax: simplify grouped expressions, evaluate powers, then perform multiplication and division before addition and subtraction.

For a negative base raised to a positive integer power, an even number of negative factors gives a positive result. An odd number leaves a negative result. That sign check applies to the actual base, so keep any separate leading minus visible.

What does a negative exponent mean?

For a nonzero real number $a$ and a positive integer $n$, the rule is:

$$a^{-n}=\frac{1}{a^n}$$

OpenStax’s integer-exponent lesson explains this reciprocal rule and why you must identify the base first. The negative exponent alone does not determine whether the answer is positive or negative.

The restriction $a\neq0$ matters. An expression such as $0^{-2}$ would require division by zero and is undefined. Also, a reciprocal can be greater than one: $\left(\frac{1}{6}\right)^{-2}=36$.

A routine for expressions with several minus signs

1. Mark the base. Include everything inside the parentheses attached to the exponent.
2. Rewrite a negative power as a reciprocal. Keep the base grouped and retain any factor outside the power.
3. Evaluate the positive power. Use even or odd parity to check a negative base’s sign.
4. Finish the remaining operations. Check the sign and simplify the fraction.

Worked example: a negative fraction as the base

Evaluate $\left(-\frac{2}{5}\right)^{-2}$. The entire fraction is the base, so its reciprocal is $-\frac{5}{2}$. The exponent becomes positive:

$$\left(-\frac{2}{5}\right)^{-2}=\left(-\frac{5}{2}\right)^2=\frac{25}{4}$$

Check by squaring the original base first: $\left(-\frac{2}{5}\right)^2=\frac{4}{25}$. Its reciprocal is $\frac{25}{4}$, and their product is $1$. A negative exponent has not made this answer negative.

Worked example: a coefficient outside the power

Evaluate $-4x^{-2}$ when $x=-3$. Substitute with parentheses around the negative value. Only $x$ carries the exponent, so the coefficient $-4$ stays outside:

$$-4(-3)^{-2}=-4\times\frac{1}{(-3)^2}=-\frac{4}{9}$$

If the expression were $(-4x)^{-2}$, the grouping would change the base. At the same value of $x$, that expression would be $12^{-2}=\frac{1}{144}$. Adding parentheses around a coefficient changes the question.

Practice: explain the sign before calculating

Try these on paper before reading the answers. For each one, identify the base and predict whether the final result is positive, negative, or zero.

1. Evaluate $-7^2+(-7)^2$.
2. Evaluate $(-3)^{-3}$.
3. Evaluate $-\left(\frac{3}{4}\right)^{-2}$.
4. Evaluate $2t^{-2}-1$ when $t=-4$.

Answers and mistake checks

1. $-7^2+(-7)^2=-49+49=0$. The first minus is outside the square; the second is inside the base. If you got $98$ or $-98$, check which minus each exponent includes. Here the two terms cancel, so the final result is zero.
2. $(-3)^{-3}=\frac{1}{(-3)^3}=-\frac{1}{27}$. The reciprocal retains the sign of the negative denominator. An answer of $-27$ misses the reciprocal step.
3. $-\left(\frac{3}{4}\right)^{-2}=-\left(\frac{4}{3}\right)^2=-\frac{16}{9}$. The leading minus remains outside the positive power. Keep it when you invert the fraction.
4. $2(-4)^{-2}-1=\frac{2}{16}-1=\frac{1}{8}-1=-\frac{7}{8}$. The exponent belongs to $t$, so only the substituted base goes into the reciprocal. The coefficient and the subtraction stay outside.

Use a missed answer to choose the next step

If you lost the sign, rewrite the expression with its base clearly grouped. If you forgot the reciprocal, write the positive-power denominator before multiplying anything else. Record that specific mistake in Kibo’s practice error-log template, then retry a fresh example with the explanation covered.

For the difference between integers and rational values such as the fractions here, see our ACT number-types guide. You can also try a free ACT practice question with Kibo.

Prepared with AI assistance. Original examples were verified with separate exact calculations; no human review is claimed. Sources checked October 12, 2026. ACT is a registered trademark of ACT, Inc., which is not affiliated with or endorsing this guide.