Zero and Negative Exponents
A negative exponent does not make a number negative. It tells you to divide. Here is what 2⁰ and 2⁻³ mean.
What you'll be able to do
- Explain why a number to the power 0 is 1.
- Rewrite a negative exponent as a fraction.
- Simplify terms with zero and negative exponents.
- Solve for a negative exponent.
- Spot common mistakes.
The big idea
Count down. 2³ = 8, 2² = 4 and 2¹ = 2. Each step down divides by 2. One more step gives 2⁰ = 1. Then 2⁻¹ = 1/2, 2⁻² = 1/4 and 2⁻³ = 1/8.
Zero exponent. Any number except 0, raised to the power 0, equals 1. So 7⁰ = 1 and x⁰ = 1.
Negative exponent. A negative exponent means 1 over the power. So 5⁻² = 1/5² = 1/25. The answer is not negative.
Only what it touches. In 2x⁻³, the exponent belongs only to x. So 2x⁻³ = 2/x³. But (2x)⁻³ = 1/(8x³), because the whole 2x is cubed.
Fractions flip. A negative power of a fraction flips it. So (2/3)⁻² = (3/2)² = 9/4.
The old rules still work. Add exponents to multiply. Take them away to divide. Watch the signs: x⁴ ÷ x⁻³ = x⁷, because 4 − (−3) = 7.
Positive form. To finish, move each negative power below the line. Then its exponent is positive. So 3x⁻⁴y² = 3y²/x⁴.
Adding is different. These rules are not for adding. To add, find each value first. 3⁻¹ + 3⁻¹ = 1/3 + 1/3 = 2/3.
Words you need
- Exponent
- The small raised number. In 5⁻², the exponent is −2.
- Base
- The number or letter being multiplied. In 5⁻², the base is 5.
- Power
- A base with an exponent, like 5⁻².
- Reciprocal
- 1 divided by a number. The reciprocal of 8 is 1/8. The reciprocal of 2/3 is 3/2.
- Coefficient
- The number in front of a letter. In 6a⁻², the coefficient is 6.
- Simplify
- Write something in its shortest form, with each letter written once.
- Square root and cube root
- The square root of 9 is 3, because 3 × 3 = 9. The cube root of 8 is 2, because 2 × 2 × 2 = 8.
The pattern
| Power | Value |
|---|---|
| 2³ | 8 |
| 2² | 4 |
| 2¹ | 2 |
| 2⁰ | 1 |
| 2⁻¹ | 1/2 |
| 2⁻² | 1/4 |
| 2⁻³ | 1/8 |
Worked examples
Example 1
Find the value of 4⁰ + 2⁻³.
- 4⁰ = 1, because any number except 0 to the power 0 is 1.
- 2⁻³ = 1/2³ = 1/8.
- 1 + 1/8 = 9/8.
Answer: 9/8
Example 2
Simplify x³ · x⁻⁵. Write your answer with positive exponents.
- You are multiplying, so add the exponents: 3 + (−5) = −2.
- That gives x⁻².
- A negative exponent means 1 over the power. So x⁻² = 1/x².
Answer: 1/x²
Example 3
Simplify 12a⁻³b⁵ ÷ (4a²b⁻¹). Write your answer with positive exponents.
- Divide the numbers: 12 ÷ 4 = 3.
- Subtract the exponents on a: −3 − 2 = −5.
- Subtract the exponents on b: 5 − (−1) = 6.
- a⁻⁵ moves below the line. So the answer is 3b⁶/a⁵.
Answer: 3b⁶/a⁵
Common mistakes
Making the answer negative: 3⁻² = −9. A negative exponent means 1 over the power. So 3⁻² = 1/9.
Saying 5⁰ = 0. Follow the pattern: 5¹ = 5, and one more step divides by 5. So 5⁰ = 1.
Moving the number with the power: 2x⁻³ = 1/(2x³). The exponent belongs only to x. So 2x⁻³ = 2/x³.
Not flipping a fraction: (2/3)⁻² = 4/9. A negative power flips the fraction first. So (2/3)⁻² = 9/4.
Practice: 20 questions
5 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Simplify 7x⁰y³.
- Write 4⁻³ as a fraction.
- Write 6a⁻²b⁴ with positive exponents.
- A micrometer is 10⁻⁶ meters. A centimeter is 10⁻² meters. How many micrometers are in a centimeter? Write your answer as a power of 10. Word problem
- Ravi says 5⁻² = −25. Find his mistake. What is the correct value?
Tier 2 Intermediate One more step to think through
- Simplify 15x⁵y⁰ ÷ (3x²y⁻³).
- Simplify (4x⁻²y³)² ÷ (2xy⁻⁴). Write your answer with positive exponents.
- Which is greater, 2 · 3⁻² or (2 · 3)⁻²?
- Find the value of (2/3)⁻².
- A ruler is marked in sixteenths of an inch. Write 1/16 of an inch as a power of 2. Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Simplify (−2x³y⁻²)⁻³ ÷ (4x⁻⁴y⁵). Write your answer with positive exponents.
- (2xᵏy⁻¹)⁻² · 4x⁶ = x⁻⁴y². What is k?
- A sheet of paper is cut in half again and again. One small piece is 1/32 of the sheet. Write 1/32 as 2ⁿ. What is n? Word problem
- A ball is dropped from 2⁵ feet. Each bounce reaches half the height of the one before. After how many bounces is it 2⁻² feet high? Word problem
- Mia says 2⁻¹ + 2⁻¹ = 2⁻². Find her mistake. What is the correct value?
Tier 4 Challenge Stretch problems, some above grade level
- Put 2⁻³⁰, 3⁻²⁰ and 5⁻¹⁰ in order from least to greatest. Do it without working them out.
- One pipe fills a tank in 2 hours. Another fills it in 3 hours. Together they take (2⁻¹ + 3⁻¹)⁻¹ hours. How long is that? Word problem
- Find the value of (3⁻⁴ + 3⁻⁴ + 3⁻⁴) ÷ 3⁻⁶.
- Solve 9ˣ⁺¹ = 27⁻¹.
- Raising a number to the power 1/3 takes its cube root. For example, 27^(1/3) = 3. Simplify (8x⁶y⁻³)^(−2/3). Write your answer with positive exponents. Above grade level
Answers
Tier 1 Foundations
- 7y³. Any number except 0, raised to the power 0, equals 1. So 7x⁰y³ = 7y³.
- 1/64. A negative exponent means 1 divided by the power. For example, 2⁻³ = 1/2³. So 4⁻³ = 1/64.
- 6b⁴/a². A negative exponent means 1 over the power, so a moves below the line. So 6a⁻²b⁴ = 6b⁴/a².
- 10⁴ = 10,000. Subtract the exponents: (−2) − (−6) = 4. So 10⁻² ÷ 10⁻⁶ = 10⁴ = 10,000.
- 1/25. Ravi made the answer negative. A negative exponent means 1 divided by the power. For example, 2⁻³ = 1/2³. So 5⁻² = 1/25.
Tier 2 Intermediate
- 5x³y³. Numbers: 15 ÷ 3 = 5. Add the exponents you multiply and subtract the ones you divide. x: 5 − 2 = 3. y: 0 − (−3) = 3. So 15x⁵y⁰ ÷ (3x²y⁻³) = 5x³y³.
- 8y¹⁰/x⁵. The power 2 applies to every part of (4x⁻²y³)²: 4² = 16, x: (−2) × 2 = −4, y: 3 × 2 = 6. So (4x⁻²y³)² = 16y⁶/x⁴. Numbers: 16 ÷ 2 = 8. Add the exponents you multiply and subtract the ones you divide. x: −4 − 1 = −5. y: 6 − (−4) = 10. A negative exponent means 1 over the power, so x moves below the line. So (4x⁻²y³)² ÷ (2xy⁻⁴) = 8y¹⁰/x⁵.
- 2 × 3⁻² is greater. 2 × 3⁻² = 2/9, and (2 × 3)⁻² = 1/36.
- 9/4. A negative exponent means 1 divided by the power. For example, 2⁻³ = 1/2³. So (2/3)⁻² = 9/4.
- 2⁻⁴ = 1/16. 16 = 2⁴, so 1/16 = 1/2⁴ = 2⁻⁴.
Tier 3 Advanced
- −y/(32x⁵). The power −3 applies to every part of (−2x³y⁻²)⁻³: (−2)⁻³ = −1/8, x: 3 × (−3) = −9, y: (−2) × (−3) = 6. So (−2x³y⁻²)⁻³ = −y⁶/(8x⁹). Numbers: (−1/8) ÷ 4 = −1/32. Add the exponents you multiply and subtract the ones you divide. x: −9 − (−4) = −5. y: 6 − 5 = 1. A negative exponent means 1 over the power, so x moves below the line. So (−2x³y⁻²)⁻³ ÷ (4x⁻⁴y⁵) = −y/(32x⁵).
- k = 5. Look at the x exponents only. (xᵏ)⁻² = x⁻²ᵏ. Multiplying by x⁶ gives x⁶⁻²ᵏ. So 6 − 2k = −4, and k = 5.
- n = −5. 32 = 2⁵, so 1/32 = 1/2⁵ = 2⁻⁵. That means n = −5. Check n = −5: left side 1/32, right side 1/32.
- n = 7. Each bounce divides the height by 2. After n bounces the height is 2⁵ ÷ 2ⁿ = 2⁵⁻ⁿ. So 5 − n = −2, and n = 7.
- 1. Mia added the exponents. That rule is only for multiplying. Here 1/2 + 1/2 = 1. Find the value of each power first. Then add or subtract. So 2⁻¹ + 2⁻¹ = 1.
Tier 4 Challenge
- 3⁻²⁰ < 2⁻³⁰ < 5⁻¹⁰. Give them all the exponent −10. 2⁻³⁰ = 8⁻¹⁰ and 3⁻²⁰ = 9⁻¹⁰. These are 1/9¹⁰, 1/8¹⁰ and 1/5¹⁰. A bigger number on the bottom makes a smaller value.
- 6/5. First add: 1/2 + 1/3 = 5/6. Then the power −1 flips it to 6/5. That is 6/5 hours, or 1 hour and 12 minutes.
- 27. Three copies of 3⁻⁴ make 3 × 3⁻⁴ = 3⁻³. Then 3⁻³ ÷ 3⁻⁶ = 3³ = 27.
- x = −5/2. 9 = 3² and 27 = 3³. So 3²ˣ⁺² = 3⁻³. Then 2x + 2 = −3, so x = −5/2.
- y²/(4x⁴). The power −2/3 applies to every part of (8x⁶y⁻³)^(−2/3): 8^(−2/3) = 1/4, x: 6 × (−2/3) = −4, y: (−3) × (−2/3) = 2. So (8x⁶y⁻³)^(−2/3) = y²/(4x⁴).
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