Why can't the base of a negative exponent be zero?

Because evaluating a negative exponent requires putting the base into the denominator of a fraction ($1 / 0^n$). Since dividing by zero is undefined in mathematics, a base of zero with a negative exponent is also undefined.

Does a negative exponent always create a fraction?

Not always. If your base is already a fraction, applying a negative exponent will flip the fraction. For example, $(1/5)^{-2}$ flips to become $5^2$, which is just the whole number 25.

What happens if both the base and the exponent are negative?

You apply the exact same rules, but be careful with your signs! The negative exponent still just flips the base. For example, $(-3)^{-2}$ becomes $1 / (-3)^2$. Then you evaluate the bottom: $(-3) \cdot (-3) = 9$. So the final answer is $1 / 9$.

How do negative exponents work?

A negative exponent works as the opposite of a positive exponent. While a positive exponent tells you how many times to multiply a base by itself, a negative exponent tells you how many times to divide by that base. Think of it like a set of stairs: if positive exponents tell you to walk up, negative exponents tell you to walk down. To calculate a negative exponent, you simply take the reciprocal of the base (flip it into a fraction) and change the exponent to a positive number. For example, $x^{-n}$ is exactly the same as $1 / x^n$ .

What is a negative exponent?

A negative exponent is simply a mathematical shorthand for division. The precise definition states that for any non-zero number $a$ and any integer $n$ , the expression $a^{-n}$ is equal to $1 / a^n$ . This means that instead of making the number larger through repeated multiplication, you are making it smaller through repeated division by moving the base into the denominator of a fraction.

Why does it work this way?

You can see why negative exponents work by looking at the rules for dividing exponents. When you divide identical bases, you subtract their exponents: $x^3 / x^5 = x^{3-5} = x^{-2}$ . If you write that same problem out as a fraction, you get $(x \cdot x \cdot x) / (x \cdot x \cdot x \cdot x \cdot x)$ . Cancel out the common terms on the top and bottom, and you are left with $1 / (x \cdot x)$ , which is $1 / x^2$ . Therefore, $x^{-2}$ must equal $1 / x^2$ .

How to recognize and evaluate them

Whenever you see a negative sign attached to an exponent, immediately think 'reciprocal.' If the negative exponent is in the numerator, move the base to the denominator and make the exponent positive. If the negative exponent is already in the denominator, like $1 / x^{-3}$ , you do the exact reverse: move the base up to the numerator to make the exponent positive, resulting in $x^3$ .

Where students slip up

The most common mistake students make is confusing a negative exponent with a negative number. Seeing a negative sign often tricks our brains into thinking the final answer will be negative. However, a negative exponent only changes the position of the base (flipping it into a fraction). It has no power to make a positive base negative. For example, $2^{-3}$ is $1 / 8$ , not $-8$ .

Worked through

Evaluate the expression $4^{-3}$ without using a calculator.

First, recognize the negative exponent. We apply the rule $a^{-n} = 1 / a^n$ . Step 1: Take the reciprocal of the base (4) and make the exponent positive. This gives us $1 / 4^3$ . Step 2: Evaluate the positive exponent in the denominator. Multiply 4 by itself three times: $4 \cdot 4 \cdot 4 = 64$ . Step 3: Write the final fraction. The answer is $1 / 64$ .

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Where this comes from: OpenStax Intermediate Algebra · Khan Academy Algebra 1 Unit: Exponents & Radicals