EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Zero, negative and fractional indices
Number · Lesson 5 of 7
Teacher copy - includes the notes for whoever is teaching from it.
Last Lesson
Answer each one, then check.
1. Write 2⁵ × 2³ as a single power of 2.
2⁸
2. Write 3⁷ ÷ 3² as a single power of 3.
3⁵
3. Write (5²)³ as a single power of 5.
5⁶
4. What is ∛27?
3
5. What is the reciprocal of 4?
¼ - one divided by the number.
Learning Objectives
1. Know that any number (except 0) to the power 0 is 1.
2. Work out negative powers as reciprocals.
3. Work out negative powers of fractions.
4. Work out fractional powers as roots. (Higher)
5. Work out powers such as 27^(−2/3). (Higher)
Follow the Pattern
Each step down, the index goes down by 1 and the value is divided by 2.
|
Power of 2 |
Value |
How it continues |
|---|---|---|
|
2³ |
8 |
|
|
2² |
4 |
8 ÷ 2 |
|
2¹ |
2 |
4 ÷ 2 |
|
2⁰ |
1 |
2 ÷ 2 |
|
2⁻¹ |
½ |
1 ÷ 2 |
|
2⁻² |
¼ |
½ ÷ 2 |
|
2⁻³ |
⅛ |
¼ ÷ 2 |
The Zero Index
Any number except 0 raised to the power 0 is 1.
▸ Using the laws. 5³ ÷ 5³ = 5³⁻³ = 5⁰. But anything divided by itself is 1, so 5⁰ = 1.
▸ Examples. 7⁰ = 1, 1000⁰ = 1, (−3)⁰ = 1, (⅔)⁰ = 1.
▸ Careful. 3 × 4⁰ = 3 × 1 = 3, not 1.
Negative Indices
A negative index means "one over": a⁻ⁿ = 1/aⁿ.
▸ Whole numbers. 4⁻² = 1/4² = 1/16. 10⁻³ = 1/1000 = 0.001.
▸ It is not negative. 2⁻³ = ⅛, a small positive number, not −8.
▸ Fractions flip. (⅔)⁻² = (3/2)² = 9/4. A negative power turns a fraction upside down.
▸ Using the laws. 2³ ÷ 2⁵ = 2⁻², and writing it out: (2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = ¼.
Negative Powers
|
Work out (a) 4⁻³ (b) (3/5)⁻² |
1. (a) Negative power: one over
4⁻³ = 1/4³
2. Work out the power
4³ = 64, so 4⁻³ = 1/64
3. (b) Negative power of a fraction: flip it
(3/5)⁻² = (5/3)²
4. Square the top and the bottom
5²/3² = 25/9
Answer: (a) 1/64 (b) 25/9 = 27/9
PART TWO · HIGHER
Fractional Indices
Powers that are roots. (The printed slides write 8 to the power two-thirds as 8^(2/3).)
Powers That Are Roots
A power of one-half is a square root; a power of one-third is a cube root.
▸ Why. 9^(1/2) × 9^(1/2) = 9¹ = 9. The number that multiplies by itself to give 9 is 3, so 9^(1/2) = √9 = 3.
▸ Unit fractions. a^(1/2) = √a, a^(1/3) = ∛a, and a^(1/n) = ⁿ√a.
▸ Examples. 25^(1/2) = 5, 64^(1/3) = 4, 81^(1/4) = 3, 32^(1/5) = 2.
▸ Other fractions. a^(m/n) = (ⁿ√a)^m: the bottom is the root, the top is the power. 8^(2/3) = (∛8)² = 2² = 4.
Three Steps for a Negative Fractional Power
Work out 27^(−2/3) one part of the index at a time.
|
1 Minus: flip 27^(−2/3) = 1/(27^(2/3)) |
2 Bottom: root The 3 means cube root: ∛27 = 3 |
3 Top: power The 2 means square: 3² = 9 |
4 Answer 27^(−2/3) = 1/9 |
A Fraction to a Fractional Power
|
Work out (16/81)^(−3/4) |
1. Minus: flip the fraction
(81/16)^(3/4)
2. Bottom of the index, 4: fourth root of the top and the bottom
∜81 = 3, ∜16 = 2
3. So far
(3/2)³
4. Top of the index, 3: cube
3³/2³ = 27/8
Answer: 27/8 = 3⅜
Solving with Powers
|
Find x if 8^x = ¼ |
1. Write both sides as powers of 2
8 = 2³ and ¼ = 2⁻²
2. So
(2³)^x = 2⁻², that is 2^(3x) = 2⁻²
3. The bases match, so the indices are equal
3x = −2
4. Divide by 3
x = −⅔
Answer: x = −⅔
Match the Power to Its Value
|
Power |
Value |
|---|---|
|
5⁰ 1 |
|
|
2⁻³ ⅛ |
|
|
49^(1/2) 7 |
|
|
8^(2/3) 4 |
|
|
1000^(1/3) 10 |
|
|
(⅓)⁻² 9 |
Case Study
|
CASE STUDY Carbon Dating and Negative Powers Living things take in a little radioactive carbon-14 while they are alive. After they die it decays, and every 5730 years - its half-life - half of what is left disappears. After 1 half-life, 2⁻¹ = ½ of the carbon-14 remains; after 2, 2⁻² = ¼; after n half-lives, 2⁻ⁿ. By measuring how much is left, scientists can work out how long ago something died. A sample with ⅛ = 2⁻³ of its carbon-14 left is about 3 × 5730 = 17 190 years old. |
|
5730 Years: the half-life of carbon-14 |
2⁻³ = ⅛ What is left after 3 half-lives (about 17 190 years) |
Key Terms
|
Zero index Any non-zero number to the power 0 equals 1. |
Negative index a⁻ⁿ = 1/aⁿ: one over the positive power. |
|
Reciprocal One divided by a number. The reciprocal of 4 is ¼; of ⅔ is 3/2. |
Fractional index (Higher) A power that is a fraction: the bottom is a root, the top is a power. |
|
nth root The number that, raised to the power n, gives the original number. |
|
Your Task: Index Sort
10 minutes
|
Sort these into three groups - "less than 1", "equal to 1" and "more than 1" - then work out each value. 3⁰, 5⁻¹, (½)⁻², 10⁻², (¾)⁰, (2/5)⁻¹. Higher: add 16^(1/2), 8^(−1/3), 27^(2/3) and (¼)^(−1/2). 1. Predict the group first. 2. Then work out the value. 3. Explain one surprise. |
A good answer shows: Less than 1: 5⁻¹ = ⅕, 10⁻² = 1/100, and (Higher) 8^(−1/3) = ½. Equal to 1: 3⁰, (¾)⁰. More than 1: (½)⁻² = 4, (2/5)⁻¹ = 5/2, and (Higher) 16^(1/2) = 4, 27^(2/3) = 9, (¼)^(−1/2) = 2.
Note: Ask why a negative power of a fraction less than 1 is always more than 1.
Can I...?
☐ Explain why a⁰ = 1.
☐ Work out a negative power of a whole number.
☐ Work out a negative power of a fraction.
☐ Write 1/aⁿ as a negative power.
☐ Work out a^(1/n) as a root. (Higher)
☐ Work out a^(m/n). (Higher)
☐ Work out negative fractional powers. (Higher)
☐ Solve equations such as 8^x = ¼. (Higher)
Summary
✓ a⁰ = 1 for any a except 0.
✓ a⁻ⁿ = 1/aⁿ; a negative power of a fraction flips it.
✓ (Higher) a^(1/n) = ⁿ√a.
✓ (Higher) a^(m/n): root with the bottom, power with the top; a minus sign flips.
|
EXAM FOCUS Find the value of 125^(−2/3) (2 marks) Do the three parts in this order - minus (flip), bottom (root), top (power) - and write each step. Taking the root before the power keeps the numbers small. |
Exam Practice: Zero, Negative and Fractional Indices
Answer all questions. Show your working. All questions are non-calculator. · 20 minutes
▸ Question 1 · 1 mark · Non-calculator. Write down the value of 7⁰.
▸ Question 2 · 1 mark · Non-calculator. Write 4⁻² as a fraction.
▸ Question 3 · 1 mark · Non-calculator · Higher. Write down the value of 64^(1/2).
▸ Question 4 · 2 marks · Non-calculator · Higher. Find the value of 125^(−2/3).
▸ Question 5 · 2 marks · Non-calculator · Higher. Find the value of (27/8)^(−2/3).
▸ Question 6 · 3 marks · Non-calculator · Higher. Write √8 × 4⁻³ as a single power of 2.
Question 1 · 1 mark · Non-calculator
|
“Write down the value of 7⁰.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 1 mark, so plan before writing.
Question 1 · mark scheme
1 mark available. Award a mark for each point made.
▸ 1. B1
▸ Model answer. 1
Question 2 · 1 mark · Non-calculator
|
“Write 4⁻² as a fraction.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 1 mark, so plan before writing.
Question 2 · mark scheme
1 mark available. Award a mark for each point made.
▸ 1/16. B1
▸ Model answer. 1/16
Question 3 · 1 mark · Non-calculator · Higher
|
“Write down the value of 64^(1/2).” |
HOW TO ANSWER IT Command word: Non-calculator · Higher. Worth 1 mark, so plan before writing.
Question 3 · mark scheme
1 mark available. Award a mark for each point made.
▸ 8. B1
▸ Model answer. 8
Question 4 · 2 marks · Non-calculator · Higher
|
“Find the value of 125^(−2/3).” |
HOW TO ANSWER IT Command word: Non-calculator · Higher. Worth 2 marks, so plan before writing.
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ 1/(125^(2/3)), or ∛125 = 5, or 25 seen. M1
▸ 1/25. A1
▸ Model answer. 125^(−2/3) = 1/(125^(2/3)). ∛125 = 5 and 5² = 25, so the answer is 1/25.
Question 5 · 2 marks · Non-calculator · Higher
|
“Find the value of (27/8)^(−2/3).” |
HOW TO ANSWER IT Command word: Non-calculator · Higher. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ (8/27)^(2/3), or 3/2 or ⅔ seen. M1
▸ 4/9. A1
▸ Model answer. Flip: (8/27)^(2/3). Cube root: ⅔. Square: 4/9.
Question 6 · 3 marks · Non-calculator · Higher
|
“Write √8 × 4⁻³ as a single power of 2.” |
HOW TO ANSWER IT Command word: Non-calculator · Higher. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ √8 written as 2^(3/2). M1
▸ 4⁻³ written as 2⁻⁶. M1
▸ 2^(−9/2). A1
▸ Model answer. √8 = 8^(1/2) = (2³)^(1/2) = 2^(3/2). 4⁻³ = (2²)⁻³ = 2⁻⁶. 2^(3/2) × 2⁻⁶ = 2^(3/2 − 6) = 2^(−9/2).