Evaluating a negative exponent in exponential notation

Key concept: This problem checks whether you can rewrite a negative exponent as a reciprocal and then simplify the power correctly. That skill shows up often in algebra, scientific notation, and function notation.

Step by step

Key idea

A negative exponent does not make a number negative. It tells you to take the reciprocal of the base and then apply the positive exponent.

For any nonzero number $a$, the rule is:

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

So the first step is always to identify the base and change the sign of the exponent.

Step-by-step method

1. Rewrite the expression using the reciprocal rule.
2. Compute the positive power.
3. Simplify the fraction.

If the intended expression is something like $9^{-2}$, then:

$9^{-2} = \frac{1}{9^2} = \frac{1}{81}$

That is why answer choices for negative-exponent questions often include a reciprocal such as $\frac{1}{81}$.

Common property to remember

A negative exponent means the base moves across the fraction bar:

• $a^{-1} = \frac{1}{a}$
• $a^{-2} = \frac{1}{a^2}$
• $a^{-3} = \frac{1}{a^3}$

This is especially useful when the base is a whole number, a variable, or a product.

Common mistakes

A frequent error is to think the exponent becomes negative in the answer, or to square the exponent itself instead of the base. For example, $9^{-2}$ is not $-18$ and it is not $9^2$.

Another mistake is forgetting that the reciprocal rule only works when the base is nonzero.

Final check

If your answer choice list includes $\frac{1}{81}$, that corresponds to a base of $9$ with exponent $-2$. The core skill being tested is still the same: convert the negative exponent into a reciprocal and simplify.

Pitfall alert

Students often read a negative exponent as a sign on the number, so they choose a negative answer like $-18$ or try to multiply the base by the exponent. That is not how exponent rules work. The exponent tells you how many times the base is used in multiplication, and a negative sign only changes the placement of the quantity: it moves to the denominator. Another trap is forgetting that the reciprocal rule applies to the entire base, not just part of it. If the base is a product or a fraction, the whole expression must be inverted before simplifying.

Try different conditions

If the question were changed to evaluate $x^{-2}$ for $x=5$, the same rule gives $5^{-2}=\frac{1}{25}$. If it were $\left(\frac{3}{4}\right)^{-2}$, you would invert the fraction first and get $\left(\frac{4}{3}\right)^2=\frac{16}{9}$. The structure of the problem stays the same; only the base changes. That is why recognizing the reciprocal rule is more important than memorizing one answer choice.

Further reading

negative exponent rule, reciprocal of a power, law of exponents