Guides / Exponents

Negative and zero exponents: what they mean and the mistakes to avoid

x⁰ = 1 and x⁻ⁿ = 1/xⁿ are two rules students memorize and then misapply. Here is why they are true, which makes them much harder to get wrong.

Why x⁰ = 1 and x⁻ⁿ = 1/xⁿ

Look at powers of 2 going down. Each step divides by 2:

1. 2³ = 8Start.
2. 2² = 48 ÷ 2
3. 2¹ = 24 ÷ 2
4. 2⁰ = 12 ÷ 2. Not zero: dividing 2 by 2 gives 1.
5. 2⁻¹ = 121 ÷ 2
6. 2⁻² = 141/2 ÷ 2. The numbers keep shrinking but stay positive.

The pattern gives the same rule as the quotient rule does. x³/x³ is obviously 1 (anything over itself), and by the rule "subtract exponents when dividing" it is also x³⁻³ = x⁰. So x⁰ has to be 1. Likewise x²/x⁵ = x⁻³, and cancelling by hand gives 1/x³. The two answers have to agree.

The one exception: 0⁰ is left undefined in algebra, and 0⁻ⁿ would mean dividing by zero, so these rules assume the base isn't 0.

Example 1: numbers

1. 5⁻² = 15² = 125Flip, then square. Positive.
2. (−3)⁰ = 1The base is −3 because of the parentheses. Anything nonzero to the 0 is 1.
3. −3⁰ = −1Without parentheses the exponent applies to 3 only, then the minus sign is applied: −(3⁰) = −1.
4. (−2)⁻³ = 1(−2)³ = 1−8 = −18The answer is negative here only because the base is negative and the power is odd, not because the exponent is negative.

Example 2: the exponent applies only to what it touches

1. 3x⁻² = 3x²The −2 is on x, not on 3. The 3 stays in the numerator.
2. (3x)⁻² = 1(3x)² = 19x²With parentheses the exponent applies to the whole product, so the 3 gets squared and moves down too.

Example 3: negative exponents in a denominator move up

"One over" applied twice puts you back where you started, so a negative exponent on the bottom of a fraction moves the factor to the top.

1. 1x⁻² = x²1 ÷ (1/x²) = x².
2. a⁻¹b⁻³ = b³aEach factor crosses the fraction bar and its exponent's sign flips. Nothing else changes.

A fraction to a negative power flips the fraction: (3/4)⁻² = (4/3)² = 16/9.

Example 4: simplifying an expression

Write x⁻² y³x⁴ y⁻¹ with positive exponents only.

1. y³ · y¹x⁴ · x²Move every factor with a negative exponent across the bar and make its exponent positive. x⁻² goes down as x², y⁻¹ comes up as y¹.
2. y⁴x⁶Add exponents on the same base when multiplying.

Or use the quotient rule directly: x⁻²⁻⁴ = x⁻⁶ and y³⁻⁽⁻¹⁾ = y⁴, then move x⁻⁶ down. Same answer, and the 3 − (−1) step is where a sign gets dropped, so many students prefer moving factors first.

Example 5: scientific notation

4.2 × 10⁻³ means 4.2 × 1/1000 = 0.0042. The negative exponent tells you the number is small (between 0 and 1), not that it is below zero. −4.2 × 10³ is the negative number, −4200.

Common mistakes, and the exact line they happen on

1. Making the answer negative

1. 2⁻³ = −8This is the mistake. The exponent's sign says "flip," not "negate." 2⁻³ = 1/2³ = 1/8. Also wrong: −1/8. The base is positive, so the result is positive.

2. Thinking x⁰ = 0

1. 7⁰ = 0This is the mistake. Look at the dividing-down pattern: 7¹ = 7, then 7 ÷ 7 = 1. So 7⁰ = 1. The same goes for (anything nonzero)⁰, including (−5)⁰ and (x² + 1)⁰.

3. Moving the coefficient along with the variable

1. 5x⁻² = 15x²This is the mistake. Only x carries the −2. The answer is 5/x². Check with x = 1: 5 · 1⁻² = 5, and 5/1² = 5, but 1/(5 · 1²) = 1/5.

4. Squaring the sign that isn't inside the parentheses

1. −2⁴ = 16This is the mistake. −2⁴ means −(2⁴) = −16. Only (−2)⁴, with the parentheses, is 16. Calculators follow this rule too.

5. Combining exponents on different bases

1. 2³ · 3² = 6⁵This is the mistake. Exponents add only when the bases match. 2³ · 3² = 8 · 9 = 72, while 6⁵ = 7776.

A quick test value catches most of these: put in x = 2 and compute both the original expression and your simplified one with a calculator. If they disagree, one of the lines in between is wrong.

Practice

1. Evaluate 4⁻².

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   1/4² = 1/16.

2. Evaluate (−6)⁰ and −6⁰.

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   (−6)⁰ = 1. −6⁰ = −(6⁰) = −1.

3. Write 7y⁻³ with a positive exponent.

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   7/y³. The 7 stays on top.

4. Evaluate (2/5)⁻².

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   Flip the fraction, then square: (5/2)² = 25/4.

5. Simplify 1/(2x)⁻³.

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   The factor moves up and the exponent turns positive: (2x)³ = 8x³.

6. Simplify a³ b⁻²a⁻¹ b² with positive exponents.

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   Move b⁻² down and a⁻¹ up: a³ · a / (b² · b²) = a⁴/b⁴.

7. Evaluate (−2)⁻³.

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   1/(−2)³ = 1/(−8) = −1/8. Negative because the base is negative and the power is odd.