Negative and Zero Exponents
A free Pre-Algebra lesson from the “Decimals, Roots, and the Real Number System” unit, with a worked example and practice problems including step-by-step solutions.
Any nonzero number raised to the zero power equals 1. A negative exponent means take the reciprocal: a^(-n) = 1 / a^n. Combined with the product, quotient, and power rules, negative and zero exponents let you write very small numbers compactly.
What you'll learn
- Use the rule that any nonzero number to the zero power equals 1
- Evaluate negative exponents as reciprocals of positive-exponent forms
- Apply exponent rules with negative or zero exponents
Why it matters: Scientific notation for very small numbers (like a virus diameter of 1.2 x 10^(-7) meters) and exponential decay (half-life, depreciation) both depend on negative exponents.
Worked example
Problem. Evaluate 5^(-2).
- A negative exponent means take the reciprocal of the positive-exponent form.
- 5^(-2) = 1 / 5^2.
- 5^2 = 25, so the answer is 1/25.
Answer: 1/25
Practice problems
1. Evaluate 4^0.
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 4^0.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Anything (nonzero) raised to the zero power is 1.
- 4^0 = 1.
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
2. Evaluate 7^0.
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 7^0.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Same rule: anything nonzero to the 0 power is 1.
- 7^0 = 1.
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
3. Evaluate 2^(-3).
Show solution
- Warm-up: First identify exactly what the question is asking: Evaluate 2^(-3).
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Negative exponent: reciprocal of 2^3.
- 2^3 = 8, so 2^(-3) = 1/8.
- Check the result by substituting or estimating: the response should match 1/8 and make sense in the original problem.
Answer: 1/8
4. Evaluate 10^(-2).
Show solution
- Core Practice: First identify exactly what the question is asking: Evaluate 10^(-2).
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- 10^(-2) = 1 / 10^2.
- 10^2 = 100, so the answer is 1/100.
- Check the result by substituting or estimating: the response should match 1/100 and make sense in the original problem.
Answer: 1/100
5. Evaluate 3^(-1).
Show solution
- Core Practice: First identify exactly what the question is asking: Evaluate 3^(-1).
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- A negative-one exponent is just the reciprocal.
- 3^(-1) = 1/3.
- Check the result by substituting or estimating: the response should match 1/3 and make sense in the original problem.
Answer: 1/3
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