How to Solve Negative Exponents and Negative Bases? (+FREE Worksheet!)
Two concepts that confuse many algebra students are negative exponents and negative bases — and they are not the same thing. A negative exponent tells you to take a reciprocal. A negative base tells you the sign of the answer depends on whether the exponent is even or odd. This lesson keeps the two ideas clear and distinct.
What Is the Difference Between a Negative Exponent and a Negative Base?
Look carefully at the parentheses:
- \(\color{blue}{(-2)^{3}}\) — the base is negative: \(\color{blue}{-2}\)
- \(\color{blue}{-2^{3}}\) — the base is \(\color{blue}{2}\); the negative sign is not inside parentheses and does not get raised to the power
- \(\color{blue}{2^{-3}}\) — the exponent is negative; the base (2) is positive
Rules
Negative Base: even exponent → positive result
A negative number multiplied by itself an even number of times gives a positive result.
- \(\color{blue}{(-2)^{4} = (-2)(-2)(-2)(-2) = 16}\)
- \(\color{blue}{(-3)^{2} = 9}\)
Negative Base: odd exponent → negative result
A negative number multiplied by itself an odd number of times stays negative.
- \(\color{blue}{(-2)^{3} = (-2)(-2)(-2) = -8}\)
- \(\color{blue}{(-5)^{1} = -5}\)
Negative sign without parentheses
Without parentheses, only the number (not the sign) is raised to the power; the negative sign is applied last.
- \(\color{blue}{-2^{3} = -(2^{3}) = -8}\)
- \(\color{blue}{-3^{2} = -(3^{2}) = -9}\)
Negative Exponent Rule (review)
\(\color{blue}{a^{-n} = \frac{1}{a^{n}}}\) Combine with a negative base carefully:
- \(\color{blue}{(-3)^{-2} = \frac{1}{(-3)^{2}} = \frac{1}{9}}\)
Step-by-Step Summary
- Check for parentheses: if the negative sign is inside, the base is negative.
- For a negative base, count whether the exponent is even (result positive) or odd (result negative).
- Without parentheses, only the numeral is raised to the power; apply the – sign afterward.
- For a negative exponent, flip to the reciprocal: \(\color{blue}{a^{-n} = \frac{1}{a^{n}}}\).
Watch: Exponents with Negative Bases (Video Lesson)
Khan Academy explains how the sign of the base interacts with even and odd exponents:
Negative Exponents and Negative Bases – Worked Examples
Example 1: Evaluate \(\color{blue}{(-4)^{2}}\).
Negative base, even exponent → positive. \(\color{blue}{(-4)(-4) = 16}\).
Answer: \(\color{blue}{(-4)^{2} = 16}\)
Example 2: Evaluate \(\color{blue}{(-2)^{5}}\).
Negative base, odd exponent → negative. \(\color{blue}{(-2)^{5} = -32}\).
Answer: \(\color{blue}{(-2)^{5} = -32}\)
Example 3: Evaluate \(\color{blue}{-3^{2}}\).
No parentheses around the -3, so only 3 is squared: \(\color{blue}{-(3^{2}) = -9}\).
Answer: \(\color{blue}{-3^{2} = -9}\)
Example 4: Evaluate \(\color{blue}{(-3)^{-2}}\).
Negative exponent: take reciprocal. \(\color{blue}{\frac{1}{(-3)^{2}} = \frac{1}{9}}\).
Answer: \(\color{blue}{(-3)^{-2} = \frac{1}{9}}\)
More Practice: Negative Exponents Video
Khan Academy provides further examples of the negative exponent rule with both positive and negative bases:
Exercises for Negative Exponents and Negative Bases
Evaluate each expression.
- \(\color{blue}{(-4)^{2}}\)
- \(\color{blue}{(-2)^{5}}\)
- \(\color{blue}{-3^{2}}\)
- \(\color{blue}{(-1)^{6}}\)
- \(\color{blue}{(-5)^{3}}\)
- \(\color{blue}{-(2^{4})}\)
Answers
- \(\color{blue}{16}\)
- \(\color{blue}{-32}\)
- \(\color{blue}{-9}\)
- \(\color{blue}{1}\)
- \(\color{blue}{-125}\)
- \(\color{blue}{-16}\)
Free Negative Exponents and Negative Bases Worksheet
Ready to practice on your own? Download our free Negative Exponents and Negative Bases worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Negative Exponents and Negative Bases before a quiz or test.
Download Properties of Exponents Worksheet
Frequently Asked Questions
Is (−3)2 the same as -32?
No. \(\color{blue}{(-3)^{2} = 9}\) because both the -3 and the exponent are inside the parentheses. \(\color{blue}{-3^{2} = -9}\) because only \(\color{blue}{3}\) is squared; the negative sign is applied after.
Can a negative base with a negative exponent give a positive result?
Yes. \(\color{blue}{(-2)^{-2} = \frac{1}{(-2)^{2}} = \frac{1}{4}}\). The negative exponent creates a reciprocal, and the even exponent makes the denominator positive.
What is a quick way to remember the even/odd rule?
Think of pairs of negatives: each pair of \(\color{blue}{(-)(-)}\) gives a positive. An even exponent means a whole number of pairs → positive. An odd exponent leaves one unpaired negative → negative result.
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