EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Calculating with powers (indices)
Number · Lesson 4 of 7
Teacher copy - includes the notes for whoever is teaching from it.
Last Lesson and Before
Answer each one, then check.
1. What is 5²?
25
2. What is √81?
9
3. Last lesson: write 72 as a product of prime factors in index form.
2³ × 3²
4. Last lesson: find the HCF of 12 and 18.
6
5. Last lesson: find the LCM of 4 and 10.
20
Learning Objectives
1. Know the square numbers to 15² and the cubes of 1, 2, 3, 4, 5 and 10.
2. Find square roots and cube roots, including negative roots.
3. Use index notation for powers.
4. Use the index laws to multiply, divide and raise powers.
5. Use the order of operations with powers and roots, with and without a calculator.
Powers and Roots
A power (or index) tells you how many times a number is multiplied by itself.
▸ Index notation. 5³ = 5 × 5 × 5 = 125. The 5 is the base; the 3 is the index or power.
▸ Squares and square roots. 7² = 49, so √49 = 7. But the equation x² = 49 has two answers: x = 7 or x = −7, since (−7)² = 49 too.
▸ Cubes and cube roots. 4³ = 64, so ∛64 = 4. A cube root of a negative number is negative: ∛(−8) = −2.
▸ Roots undo powers. Squaring and square-rooting are inverse operations, like × and ÷.
Numbers to Know by Heart
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Square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 (up to 15²). |
Cube numbers 1, 8, 27, 64, 125 (1³ to 5³) and 10³ = 1000. |
|
Powers of 2 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 (2¹ to 2¹⁰). |
Powers of 10 10, 100, 1000, 10 000 - the index is the number of zeros. |
Why "Squared" and "Cubed"?
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A square with sides 3 units long is made of 3 × 3 = 9 unit squares - that is why 3² is called "3 squared". A cube with edges 3 units long is made of 3 × 3 × 3 = 27 unit cubes - "3 cubed". |
3² counts the squares in a square; 3³ counts the cubes in a cube. |
PART ONE
The Index Laws
Shortcuts for powers of the same number.
The Three Index Laws
They only work when the base is the same.
|
Law |
In words |
Example |
|---|---|---|
|
Multiplying |
Add the indices |
3⁴ × 3² = 3⁶ |
|
Dividing |
Subtract the indices |
5⁷ ÷ 5³ = 5⁴ |
|
Power of a power |
Multiply the indices |
(2³)⁴ = 2¹² |
|
Different bases |
The laws do not apply: work each power out |
2³ × 5² = 8 × 25 = 200 |
Why the Laws Work
Write the powers out in full and count.
▸ Multiplying. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3): six 3s multiplied, so 3⁶.
▸ Dividing. 5⁷ ÷ 5³: three of the seven 5s cancel, leaving four: 5⁴.
▸ Power of a power. (2³)⁴ = 2³ × 2³ × 2³ × 2³: four lots of three 2s, so 2¹².
▸ In letters. a^m × aⁿ = a^(m+n), a^m ÷ aⁿ = a^(m−n) and (a^m)ⁿ = a^(mn).
Using the Index Laws
|
Work out the value of (7⁵ × 7³)/7⁶ |
1. Multiplying: add the indices
7⁵ × 7³ = 7⁸
2. Dividing: subtract the indices
7⁸ ÷ 7⁶ = 7²
3. Work out the value
7² = 49
Answer: 49
Changing the Base
|
Write 4³ × 8² as a single power of 2. |
1. Write each base as a power of 2
4 = 2² and 8 = 2³
2. Power of a power: multiply
4³ = (2²)³ = 2⁶ and 8² = (2³)² = 2⁶
3. Multiplying: add
2⁶ × 2⁶ = 2¹²
4. Check
4³ × 8² = 64 × 64 = 4096 = 2¹²
Answer: 2¹²
PART TWO
Calculating with Powers
Order of operations, with and without a calculator.
Order of Operations
Powers and roots come before multiplying, dividing, adding and subtracting.
▸ BIDMAS. Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
▸ Negative numbers. −3² = −9 (square 3, then make it negative), but (−3)² = 9.
▸ A root sign is a bracket. √(9 + 16) = √25 = 5. It is NOT √9 + √16 = 7.
▸ On a calculator. Use brackets for the top and bottom of a fraction, and write down the full display before rounding.
A Non-Calculator Calculation
|
Work out 2³ × 5² − √144 |
1. Powers and roots first
2³ = 8, 5² = 25, √144 = 12
2. Then multiply
8 × 25 = 200
3. Then subtract
200 − 12 = 188
Answer: 188
Mistakes to Avoid
|
WRONG |
RIGHT |
|
▸ 3⁴ × 3² = 9⁶ ▸ 2³ = 6 ▸ 5² + 5³ = 5⁵ ▸ (−4)² = −16 |
▸ 3⁴ × 3² = 3⁶ - the base stays the same. ▸ 2³ = 2 × 2 × 2 = 8 ▸ 5² + 5³ = 25 + 125 = 150 - no law for adding. ▸ (−4)² = 16 - negative times negative is positive. |
Case Study
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CASE STUDY Rice on a Chessboard An old legend tells of a clever inventor who showed a king the game of chess. As a reward he asked for one grain of rice on the first square of the board, two on the second, four on the third, and so on, doubling each time. The king laughed at such a small request - until his treasurers did the maths. The 64th square alone needs 2⁶³ grains, and the whole board needs 2⁶⁴ − 1: about 18 quintillion grains, far more rice than has ever been grown. The story is a legend, but the numbers are real, and they show how fast powers grow. |
|
2⁶³ Grains on the last square alone |
1.8 × 10¹⁹ Grains on the whole board |
Key Terms
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Power (index) The small number that says how many times the base is multiplied by itself. |
Base The number being multiplied, e.g. the 5 in 5³. |
|
Square number A number made by multiplying a whole number by itself, e.g. 49 = 7². |
Cube number A number made by multiplying a whole number by itself three times, e.g. 64 = 4³. |
|
Square root The number that squares to give a number: √49 = 7. |
Cube root The number that cubes to give a number: ∛64 = 4. |
|
Index laws The rules for multiplying, dividing and raising powers of the same base. |
|
Your Task: Power Pyramids
10 minutes
|
Write each as a single power, then as an ordinary number. (a) 2⁵ × 2³ (b) 10⁸ ÷ 10⁵ (c) (3²)³ (d) 5⁹ ÷ (5² × 5⁴) (e) 9² × 3³ as a power of 3 (f) find n if 2ⁿ = 4⁵ ÷ 2³ 1. Use the index laws. 2. Change a base where you need to. 3. Check one answer the long way. |
A good answer shows: (a) 2⁸ = 256 (b) 10³ = 1000 (c) 3⁶ = 729 (d) 5³ = 125 (e) 3⁴ × 3³ = 3⁷ = 2187 (f) 4⁵ = 2¹⁰, so 2¹⁰ ÷ 2³ = 2⁷ and n = 7
Note: Parts (e) and (f) need a base changed first - a good stretch for students who finish (a) to (d) quickly.
Can I...?
☐ Recall square numbers to 15².
☐ Recall the cubes of 1, 2, 3, 4, 5 and 10.
☐ Find square roots and cube roots.
☐ Give both square roots of a number.
☐ Multiply powers of the same base.
☐ Divide powers of the same base.
☐ Raise a power to a power.
☐ Use BIDMAS with powers and roots.
Summary
✓ 5³ means 5 × 5 × 5; a root undoes a power.
✓ Same base: multiply, add indices; divide, subtract indices; power of a power, multiply indices.
✓ Different bases: work each power out separately.
✓ Powers and roots come before ×, ÷, + and −; a root sign acts as a bracket.
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EXAM FOCUS Show that 9⁴ × 27² = 3¹⁴ (3 marks) In a "show that" question the answer is given, so the marks are all for the working. Write every step, including 9 = 3² and 27 = 3³. |
Exam Practice: Calculating with Powers
Answer all questions. Show your working. Questions 1 to 5 are non-calculator. · 25 minutes
▸ Question 1 · 1 mark · Non-calculator. Write down the value of 3⁴.
▸ Question 2 · 2 marks · Non-calculator. Write (5⁶ × 5³)/5⁴ as a single power of 5.
▸ Question 3 · 2 marks · Non-calculator. Work out 2³ × 5² − √144
▸ Question 4 · 2 marks · Non-calculator. 2ⁿ = 4⁵ ÷ 2³. Find the value of n.
▸ Question 5 · 3 marks · Non-calculator · Show that. Show that 9⁴ × 27² = 3¹⁴
▸ Question 6 · 2 marks · Calculator. (a) Work out (2.7³ − √18.5)/1.4². Write down all the figures on your calculator display. (b) Give your answer to part (a) correct to 3…
Question 1 · 1 mark · Non-calculator
|
“Write down the value of 3⁴.” |
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.
▸ 81. B1
▸ Model answer. 81
Question 2 · 2 marks · Non-calculator
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“Write (5⁶ × 5³)/5⁴ as a single power of 5.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ 5⁹ seen, or a correct method for dividing. M1
▸ 5⁵. A1
▸ Model answer. 5⁶ × 5³ = 5⁹, and 5⁹ ÷ 5⁴ = 5⁵
Question 3 · 2 marks · Non-calculator
|
“Work out 2³ × 5² − √144” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 3 · mark scheme
2 marks available. Award a mark for each point made.
▸ Two of 8, 25 and 12 seen, or 200 seen. M1
▸ 188. A1
▸ Model answer. 8 × 25 − 12 = 200 − 12 = 188
Question 4 · 2 marks · Non-calculator
|
“2ⁿ = 4⁵ ÷ 2³. Find the value of n.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ 4⁵ written as 2¹⁰, or 1024 ÷ 8 = 128. M1
▸ n = 7. A1
▸ Model answer. 4⁵ = (2²)⁵ = 2¹⁰, so 2¹⁰ ÷ 2³ = 2⁷ and n = 7.
Question 5 · 3 marks · Non-calculator · Show that
|
“Show that 9⁴ × 27² = 3¹⁴” |
HOW TO ANSWER IT Command word: Non-calculator · Show that. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ 9 = 3² or 27 = 3³ used. M1
▸ 3⁸ and 3⁶. M1
▸ Fully correct working leading to 3¹⁴. A1
▸ Model answer. 9 = 3², so 9⁴ = (3²)⁴ = 3⁸. 27 = 3³, so 27² = (3³)² = 3⁶. 3⁸ × 3⁶ = 3¹⁴.
Question 6 · 2 marks · Calculator
|
“(a) Work out (2.7³ − √18.5)/1.4². Write down all the figures on your calculator display. (b) Give your answer to part (a) correct to 3 significant figures.” |
HOW TO ANSWER IT Command word: Calculator. Worth 2 marks, so plan before writing.
Question 6 · mark scheme
2 marks available. Award a mark for each point made.
▸ (a) 7.8478… (at least 5 figures). B1
▸ (b) 7.85, follow through from (a). B1
▸ Model answer. (a) 7.847876207 (b) 7.85