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. In Decimals, Roots, and the Real Number System, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.
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
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.
- Connect the calculation back to Negative and Zero Exponents so the method, not just the arithmetic, is clear.
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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