Does a^0 always equal 1?

Yes, for any nonzero number a. The rule is a^0 = 1. However, 0^0 is undefined — mathematicians do not assign it a value. So as long as your base is not zero, the zero exponent rule applies.

Why does 2^-3 = 1/8 and not -8?

The negative exponent tells you to put the base in the denominator and make the exponent positive. So 2^-3 = 1/2^3 = 1/8. The negative sign in the exponent does not make the answer negative. Think of it as a location instruction, not a sign instruction.

How do I simplify 5^-2 × 5^3?

Use the product rule: when multiplying powers with the same base, add the exponents. 5^-2 × 5^3 = 5^-2+3 = 5^1 = 5. You do not need to convert to fractions first — add the exponents directly, even if one is negative.

When I divide by a power with a negative exponent, what happens?

Use the quotient rule: subtract the exponents. For example, 3^4 3^-2 = 3^4-(-2) = 3^4+2 = 3^6. Be careful: subtracting a negative is the same as adding a positive. When you divide by a negative exponent, the exponent gets bigger.

M8MATH-2.2

Zero & Negative Exponents

Learn why any nonzero number to the zero power equals 1, and how negative exponents mean reciprocals. Master expressions like 2⁻³ and understand the rules through division patterns.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Zero & Negative Exponents, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know that

3^2 = 9

and

3^3 = 27

. But what happens when the exponent is zero, or negative? It might seem strange — does

3^0

equal 0, or something else? And what could

3^{-2}

possibly mean? These questions have real answers that follow directly from the division rule you've already learned. In this lesson, you'll extend what you know about exponents to discover why

a^0 = 1

for any nonzero number

a

, and why

a^{-n}

is the same as

\frac{1}{a^n}

Understanding Zero Exponents Through Division

Start with what you know: the quotient rule says that when you divide powers with the same base, you subtract the exponents. For example,

\frac{a^5}{a^2} = a^{5-2} = a^3

. Now let's use this rule with exponents that are equal. What is

\frac{a^3}{a^3}

? Using the quotient rule, you get

a^{3-3} = a^0

. But you also know that any nonzero number divided by itself equals 1:

\frac{a^3}{a^3} = 1

. Both answers must be true, so

a^0 = 1

. This works for any nonzero number:

5^0 = 1

(-7)^0 = 1

, and

\left(\frac{2}{3}\right)^0 = 1

. The key is that the base cannot be zero, because

0^0

is undefined in mathematics. Zero exponents come directly from division — they're not a special case, they're a logical consequence of the quotient rule.

Understanding Negative Exponents Through Division

Now let's look at

\frac{a^2}{a^5}

. Using the quotient rule:

\frac{a^2}{a^5} = a^{2-5} = a^{-3}

. What does this mean by dividing it out step by step? You can write

\frac{a^2}{a^5} = \frac{a \cdot a}{a \cdot a \cdot a \cdot a \cdot a}

. Cancel the two

a

's on top, and you have three

a

's left on the bottom:

\frac{1}{a^3}

. So

a^{-3} = \frac{1}{a^3}

. The negative exponent means "put this factor in the denominator and make the exponent positive." In general,

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

. This rule applies to any nonzero base. For example,

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

and

10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01

. Negative exponents do not make negative numbers — they create fractions.

Evaluating Expressions with Negative and Zero Exponents

Now you can evaluate any expression with zero or negative exponents. Start with

2^{-3}

: this means

\frac{1}{2^3} = \frac{1}{8}

. Next, try

\left(\frac{1}{2}\right)^{-2}

. Write it as

\frac{1}{\left(\frac{1}{2}\right)^2}

. The denominator is

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

, so you have

\frac{1}{\frac{1}{4}} = 1 \div \frac{1}{4} = 1 \times \frac{4}{1} = 4

. Notice: a negative exponent in the numerator flips the base to its reciprocal. If the base is already a fraction, a negative exponent flips it:

\left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^{2} = 4

. When you combine zero and negative exponents with the product rule, subtract exponents carefully. For example,

3^2 \times 3^{-5} = 3^{2+(-5)} = 3^{-3} = \frac{1}{3^3} = \frac{1}{27}

Common Misconception: Negative Exponents Make Negative Numbers

This is the biggest trap students fall into. A negative exponent does not make the result negative. The exponent tells you about where the base goes — to the denominator — not the sign of the answer. For example,

3^{-2} = \frac{1}{3^2} = \frac{1}{9}

, which is positive. And

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

, also positive, because the exponent is even. The sign of the result depends on the sign of the base and whether the exponent is even or odd — the same rule as positive exponents. If you see

-3^{-2}

, with the negative sign outside the base, this means

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

. But

(-3)^{-2}

with the negative sign in parentheses means

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

. Always look carefully at where the parentheses are.

Key terms

Zero exponent.: For any nonzero number $a$ , the expression $a^0 = 1$ . It comes from the quotient rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$ , and any number divided by itself is 1.
Negative exponent.: For any nonzero number $a$ and positive integer $n$ , the expression $a^{-n} = \frac{1}{a^n}$ . A negative exponent means the base is in the denominator with a positive exponent.
Quotient rule for exponents.: When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ (where $a \neq 0$ ).
Reciprocal.: The reciprocal of a number $a$ is $\frac{1}{a}$ . Multiplying a number by its reciprocal always gives 1.
Base.: The number being raised to a power. In the expression $5^3$ , the base is 5.
Exponent.: The number that tells how many times the base is multiplied by itself. In the expression $5^3$ , the exponent is 3.

Worked example

Evaluate

4^{-2} \times 2^3 \div 2^{-1}

Start by rewriting each term so all exponents are positive.

4^{-2} = \frac{1}{4^2} = \frac{1}{16}

. For the division

2^3 \div 2^{-1}

, use the quotient rule:

2^3 \div 2^{-1} = 2^{3-(-1)} = 2^{3+1} = 2^4 = 16

. Now the expression is

\frac{1}{16} \times 16

. This equals

\frac{16}{16} = 1

. You can also check by converting everything to positive exponents first:

4^{-2} \times 2^3 \times 2^{1} = \frac{1}{16} \times 8 \times 2 = \frac{16}{16} = 1

Practice questions

Which expression is equivalent to

5^{-3}

$-5^3$
$\frac{1}{5^3}$
$-\frac{1}{5^3}$
$5 \times (-3)$

Answer: $\frac{1}{5^3}$

By definition,

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

. So

5^{-3} = \frac{1}{5^3} = \frac{1}{125}

. The negative exponent does not make the result negative — it moves the base to the denominator. The other choices confuse exponents with multiplication or add a negative sign incorrectly.

Evaluate

\left(\frac{3}{2}\right)^0 \times 6^{-1}

. Show your work and express your answer as a fraction.

Answer: $\frac{1}{6}$

First,

\left(\frac{3}{2}\right)^0 = 1

because any nonzero number to the zero power equals 1. Then

6^{-1} = \frac{1}{6}

. So the expression is

1 \times \frac{1}{6} = \frac{1}{6}

. The zero exponent doesn't change the value — it makes the whole term equal to 1, leaving you to multiply by the second term.

Simplify

7^4 \times 7^{-6}

using the product rule for exponents. What is the answer?

$\frac{1}{49}$
$49$
$-2$
$\frac{1}{7^2}$

Answer: $\frac{1}{49}$

Using the product rule,

7^4 \times 7^{-6} = 7^{4+(-6)} = 7^{-2}

. Now

7^{-2} = \frac{1}{7^2} = \frac{1}{49}

. Both

\frac{1}{49}

and

\frac{1}{7^2}

are correct, but they are the same value. The exponent is negative, yet the result is a positive fraction — not negative.

FAQ

Does $a^0$ always equal 1?: Yes, for any nonzero number $a$ . The rule is $a^0 = 1$ . However, $0^0$ is undefined — mathematicians do not assign it a value. So as long as your base is not zero, the zero exponent rule applies.
Why does $2^{-3} = \frac{1}{8}$ and not $-8$ ?: The negative exponent tells you to put the base in the denominator and make the exponent positive. So $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ . The negative sign in the exponent does not make the answer negative. Think of it as a location instruction, not a sign instruction.
How do I simplify $5^{-2} \times 5^3$ ?: Use the product rule: when multiplying powers with the same base, add the exponents. $5^{-2} \times 5^3 = 5^{-2+3} = 5^1 = 5$ . You do not need to convert to fractions first — add the exponents directly, even if one is negative.
When I divide by a power with a negative exponent, what happens?: Use the quotient rule: subtract the exponents. For example, $3^4 \div 3^{-2} = 3^{4-(-2)} = 3^{4+2} = 3^6$ . Be careful: subtracting a negative is the same as adding a positive. When you divide by a negative exponent, the exponent gets bigger.

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