Indices
🎯 In this topic you will
- Use positive, negative, and zero indices to represent numbers
- Apply index laws for multiplication and division
🧠 Key Words
- generalise
- power
Show Definitions
- generalise: To describe a pattern or rule that applies to all similar situations, often using algebra.
- power: The number of times a base is multiplied by itself, shown as an exponent.
🔼 Exploring Powers of Numbers
In this section you will investigate numbers written as powers.
Look at these powers of 5:
| n | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 5n | 1 | 5 | 25 | 125 | 625 | 3125 |
So 53 = 5 × 5 × 5 = 125 and 54 = 5 × 5 × 5 × 5 = 625 and so on.
As you move to the right the numbers in the bottom row multiply by 5.
As you move to the left the numbers in the bottom row divide by 5.
3125 ÷ 5 = 625, 625 ÷ 5 = 125, 125 ÷ 5 = 25
If you continue to divide by 5, 25 ÷ 5 = 5 so 51 = 5
There is another number missing in the table. What is 50?
Divide by 5 again: 50 = 51 ÷ 5 = 5 ÷ 5 = 1
So 50 = 1
If n is any positive integer then n0 = 1.
❓ EXERCISES
1. Copy and complete this list of powers of 2.
| Power | $2^0$ | $2^1$ | $2^2$ | $2^3$ | $2^4$ | $2^5$ | $2^6$ | $2^7$ | $2^8$ | $2^9$ | $2^{10}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Number | 1 | 2 | 8 | 64 | 512 |
👀 Show answer
$2^1 = 2$
$2^2 = 4$
$2^3 = 8$
$2^4 = 16$
$2^5 = 32$
$2^6 = 64$
$2^7 = 128$
$2^8 = 256$
$2^9 = 512$
$2^{10} = 1024$
2. Copy and complete this list of powers of 3.
| Power | $3^0$ | $3^1$ | $3^2$ | $3^3$ | $3^4$ | $3^5$ | $3^6$ | $3^7$ | $3^8$ |
|---|---|---|---|---|---|---|---|---|---|
| Number | 3 | 27 | 2187 |
👀 Show answer
$3^1 = 3$
$3^2 = 9$
$3^3 = 27$
$3^4 = 81$
$3^5 = 243$
$3^6 = 729$
$3^7 = 2187$
$3^8 = 6561$
🧠 Think like a Mathematician
Question: What happens when you multiply numbers written as powers of 2 or powers of 3?
Equipment: Calculator (optional), pencil, paper
Look at this multiplication: $4 \times 16 = 64$
You can write all the numbers as powers of 2: $2^2 \times 2^4 = 2^6$
- a) Write each of these multiplications as powers of 2:
- $8 \times 4 = 32$
- $16 \times 8 = 128$
- $4 \times 32 = 128$
- $2 \times 128 = 256$
- $16 \times 32 = 512$
- b) Can you see a pattern in your answers? Make a conjecture about multiplying powers of 2. Test your conjecture on some more multiplications of your own.
- c) Make a conjecture about multiplying powers of 3. Use some examples to test your conjecture.
- d)Generalise your results so far.
🔎 Reasoning Tip
Generalising: Generalising means using a set of results to come up with a general rule.
Follow-up Questions:
👀 show answer
- a.i:$8 = 2^3$, $4 = 2^2$, so $2^3 \times 2^2 = 2^5 = 32$
- a.ii:$16 = 2^4$, $8 = 2^3$, so $2^4 \times 2^3 = 2^7 = 128$
- a.iii:$4 = 2^2$, $32 = 2^5$, so $2^2 \times 2^5 = 2^7 = 128$
- a.iv:$2 = 2^1$, $128 = 2^7$, so $2^1 \times 2^7 = 2^8 = 256$
- a.v:$16 = 2^4$, $32 = 2^5$, so $2^4 \times 2^5 = 2^9 = 512$
- Pattern: When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$
- Powers of 3 Example:$3^2 \times 3^3 = 3^5 = 243$
- General Rule: Multiplying powers with the same base always results in adding the exponents.
❓ EXERCISES
3. Write the answers to these calculations as powers of 6:
a) $6^2 \times 6^3$
b) $6^4 \times 6$
c) $6^5 \times 6^2$
d) $6^3 \times 6^3$
👀 Show answer
b) $6^5$
c) $6^7$
d) $6^6$
4. Write the answers to these calculations in index form:
a) $10^3 \times 10^2$
b) $20^5 \times 20$
c) $15^3 \times 15^3$
d) $5^5 \times 5^3$
👀 Show answer
b) $20^6$
c) $15^6$
d) $5^8$
5. a) $3^8 = 6561$. Use this fact to find $3^9$ and show your method.
b) $5^6 = 15\,625$. Use this fact to find $5^7$ and show your method.
👀 Show answer
b) $5^7 = 5^6 \times 5 = 15\,625 \times 5 = 78\,125$
6. Find the missing power:
a) $3^3 \times 3^{\square} = 3^5$
b) $9^3 \times 9^{\square} = 9^8$
c) $12^4 \times 12^{\square} = 12^6$
d) $15^{\square} \times 15^3 = 15^{10}$
👀 Show answer
b) Missing power = $5$ → $9^3 \times 9^5 = 9^8$
c) Missing power = $2$ → $12^4 \times 12^2 = 12^6$
d) Missing power = $7$ → $15^7 \times 15^3 = 15^{10}$
7. Read what Sofia says:
$4^2$ is equal to $2^4$ and $4^3$ is equal to $3^4$
Is Sofia correct? Give a reason for your answer.
👀 Show answer
$4^2 = 16$ and $2^4 = 16$ ✅
But $4^3 = 64$ and $3^4 = 81$ ❌
So the second statement is not true.
8. A million is $10^6$. A billion is 1000 million.
Write as a power of 10:
a) one billion
b) 1000 billion
👀 Show answer
b) $1000 \times 10^9 = 10^3 \times 10^9 = 10^{12}$
9. Write in index form:
a) $2^2 \times 2^3 \times 2$
b) $3^3 \times 3^4 \times 3^2$
c) $5 \times 5^3 \times 5^3$
d) $10^3 \times 10^2 \times 10^4$
👀 Show answer
b) $3^9$
c) $5^7$
d) $10^9$
10. a) $(3^2)^3 = 3^2 \times 3^2 \times 3^2$. Write $(3^2)^3$ as a single power of 3.
b) Write in index form:
i) $(2^3)^2$
ii) $(5^3)^2$
iii) $(4^2)^3$
iv) $(15^2)^4$
v) $(10^4)^3$
c) $N$ is a positive integer. Write in index form:
i) $(N^2)^3$
ii) $(N^4)^2$
iii) $(N^5)^3$
d) Can you generalise the results of part c?
👀 Show answer
b) i) $2^{3 \times 2} = 2^6$
ii) $5^{3 \times 2} = 5^6$
iii) $4^{2 \times 3} = 4^6$
iv) $15^{2 \times 4} = 15^8$
v) $10^{4 \times 3} = 10^{12}$
c) i) $N^6$
ii) $N^8$
iii) $N^{15}$
d) General rule: $(N^a)^b = N^{a \times b}$
🧠 Think like a Mathematician
Question: How can divisions be written using indices (powers)? What patterns can you discover?
Equipment: Pencil, paper, calculator (optional)
Here is a division: $32 \div 4 = 8$
You can write this using indices: $2^5 \div 2^2 = 2^3$
- a) Write each of these divisions using indices. All the numbers are powers of 2 or 3:
- $64 \div 4 = 16$
- $81 \div 3 = 27$
- $512 \div 16 = 32$
- $729 \div 9 = 81$
- $9 \div 9 = 1$
- b) Write some similar divisions using powers of 5.
- c) Can you generalise your results from a and b? Check with some powers of other positive integers.
- d) Compare your results with a partner’s.
Follow-up Questions:
👀 show answer
- a.i:$64 = 2^6$, $4 = 2^2$, so $2^6 \div 2^2 = 2^4 = 16$
- a.ii:$81 = 3^4$, $3 = 3^1$, so $3^4 \div 3^1 = 3^3 = 27$
- a.iii:$512 = 2^9$, $16 = 2^4$, so $2^9 \div 2^4 = 2^5 = 32$
- a.iv:$729 = 3^6$, $9 = 3^2$, so $3^6 \div 3^2 = 3^4 = 81$
- a.v:$9 = 3^2$, $9 = 3^2$, so $3^2 \div 3^2 = 3^0 = 1$
- General Rule:$a^m \div a^n = a^{m-n}$
❓ EXERCISES
11. Write the answers to these calculations in index form:
a) $2^7 \div 2^5$
b) $10^6 \div 10^3$
c) $15^8 \div 15^6$
d) $8^{10} \div 8^9$
e) $2^{15} \div 2^{11}$
f) $2^5 \div 2^5$
👀 Show answer
b) $10^3$
c) $15^2$
d) $8^1 = 8$
e) $2^4$
f) $2^0 = 1$
12. Write the answers to these calculations in index form:
a) $9^5 \times 9^2$
b) $9^5 \div 9^2$
c) $(9^5)^2$
d) $5^5 \times 5^4$
e) $12^8 \div 12^3$
f) $(7^3)^3$
g) $(10^1)^4$
👀 Show answer
b) $9^3$
c) $9^{10}$
d) $5^9$
e) $12^5$
f) $7^9$
g) $10^4$
13. Read what Zara says:
"I think that $(5^2)^3 = (5^3)^2$"
a) Is Zara correct? Give a reason for your answer.
b) Is a similar result true for other indices?
👀 Show answer
$(5^2)^3 = 5^{2 \times 3} = 5^6$
$(5^3)^2 = 5^{3 \times 2} = 5^6$
Although the results are equal, it is **not** because the expressions are equal in structure — the order matters.
The equality holds because $2 \times 3 = 3 \times 2$, but $(a^b)^c$ is only equal to $(a^c)^b$ **if $bc = cb$**, which is true numerically but not algebraically in all cases.
b) Yes, this is true for other indices where $m \times n = n \times m$ (which is always the case for real numbers).
14. $15 = 3 \times 5$
Use this fact to write as a product of prime factors:
a) $15^2$
b) $15^3$
c) $15^5$
d) $15^8$
👀 Show answer
b) $15^3 = (3 \times 5)^3 = 3^3 \times 5^3$
c) $15^5 = 3^5 \times 5^5$
d) $15^8 = 3^8 \times 5^8$
15. a) Write $5^6 \div 5^4$ as a power of 5.
b) Write $5^6 \div 5^6$ as a power of 5.
c) Is it possible to write $5^4 \div 5^6$ as a power of 5?
👀 Show answer
b) $5^6 \div 5^6 = 5^0 = 1$
c) Yes, $5^4 \div 5^6 = 5^{4 - 6} = 5^{-2}$
🍬 Learning Bridge
Now that you’ve mastered powers with positive integers — including how to multiply and divide them using index laws — you're ready to extend these ideas to zero and negative powers. You'll discover why \(a^0 = 1\), what negative indices mean, and how they fit perfectly into the same patterns you've already seen.
🔢 Table of Powers of 3
This table shows powers of 3.
| 32 | 33 | 34 | 35 | 36 |
| 9 | 27 | 81 | 243 | 729 |
🔎 Reasoning Tip
Index notation: The index is the small red number written above and to the right of the base.
🔁 Extending Powers of 3
When you move one column to the right, the index increases by 1 and the number multiplies by 3.
9 × 3 = 27 27 × 3 = 81 81 × 3 = 243, and so on.
When you move one column to the left, the index decreases by 1 and the number divides by 3. You can use this fact to extend the table to the left:
| 3–4 | 3–3 | 3–2 | 3–1 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1⁄81 | 1⁄27 | 1⁄9 | 1⁄3 | 1 | 3 | 9 | 27 | 81 | 243 | 729 |
🔎 Reasoning Tip
Power of zero: \( 3^0 = 1 \) may seem unusual, but it follows the pattern of dividing by 3 each time the power decreases.
📉 Understanding Negative Powers
9 ÷ 3 = 3 3 ÷ 3 = 1 1 ÷ 3 = $\dfrac{1}{3}$ $\dfrac{1}{3}$ ÷ 3 = $\dfrac{1}{9}$ $\dfrac{1}{9}$ ÷ 3 = $\dfrac{1}{27}$, and so on.
You can see from the table that 31 = 3 and 30 = 1.
Also: 3–1 = $\dfrac{1}{3}$ 3–2 = $\dfrac{1}{3^2}$ 3–3 = $\dfrac{1}{3^3}$, and so on.
In general, if n is a positive integer then 3–n = $\dfrac{1}{3^n}$. These results are not only true for powers of 3. They apply to any positive integer.
For example: 5–2 = $\dfrac{1}{5^2}$ = $\dfrac{1}{25}$ 8–3 = $\dfrac{1}{8^3}$ = $\dfrac{1}{512}$ 60 = 1
In general, if a and n are positive integers then a0 = 1 and a–n = $\dfrac{1}{a^n}$.
❓ EXERCISE
16. Write each number as a fraction:
a) $4^{-1}$
b) $2^{-3}$
c) $9^{-2}$
d) $6^{-3}$
e) $10^{-4}$
f) $2^{-5}$
👀 Show answer
b) $2^{-3} = \dfrac{1}{8}$
c) $9^{-2} = \dfrac{1}{81}$
d) $6^{-3} = \dfrac{1}{216}$
e) $10^{-4} = \dfrac{1}{10\,000}$
f) $2^{-5} = \dfrac{1}{32}$
17. Here are five numbers:
$2^{-4},\quad 3^{-3},\quad 4^{-2},\quad 5^{-1},\quad 6^0$
List the numbers in order of size, smallest first.
👀 Show answer
$3^{-3} = \dfrac{1}{27} \approx 0.037$
$2^{-4} = \dfrac{1}{16} = 0.0625$
$4^{-2} = \dfrac{1}{16} = 0.0625$
$5^{-1} = \dfrac{1}{5} = 0.2$
$6^0 = 1$
**Order (smallest to largest):**
$3^{-3},\ 2^{-4},\ 4^{-2},\ 5^{-1},\ 6^0$
18. Write these numbers as powers of 2:
a) $\dfrac{1}{2}$
b) $\dfrac{1}{4}$
c) $64$
d) $\dfrac{1}{64}$
e) $1$
f) $8^{-1}$
👀 Show answer
b) $2^{-2}$
c) $2^6$
d) $2^{-6}$
e) $2^0$
f) $2^{-3}$ (since $8 = 2^3$)
19. Write each number as a power of 10:
a) $100$
b) $1000$
c) $1$
d) $0.1$
e) $0.001$
f) $0.000001$
👀 Show answer
b) $10^3$
c) $10^0$
d) $10^{-1}$
e) $10^{-3}$
f) $10^{-6}$
20. Write $\dfrac{1}{64}$
a) as a power of 64
b) as a power of 8
c) as a power of 4
d) as a power of 2
👀 Show answer
b) $8^{-2}$
c) $4^{-3}$
d) $2^{-6}$
21. a) Write $\dfrac{1}{81}$ as a power of a positive integer.
b) How many different ways can you write the answer to part a?
👀 Show answer
b) Possible alternative forms:
→ $9^2 = 81$ → $\dfrac{1}{81} = 9^{-2}$
→ $81^1 = 81$ → $\dfrac{1}{81} = 81^{-1}$
So at least 3 different forms.
22. When $x = 6$, find the value of:
a) $x^2$
b) $x^{-2}$
c) $x^0$
d) $x^{-3}$
👀 Show answer
b) $6^{-2} = \dfrac{1}{36}$
c) $6^0 = 1$
d) $6^{-3} = \dfrac{1}{216}$
23. Write $m^{-2}$ as a fraction when:
a) $m = 9$
b) $m = 15$
c) $m = 1$
d) $m = 20$
👀 Show answer
b) $15^{-2} = \dfrac{1}{225}$
c) $1^{-2} = 1$
d) $20^{-2} = \dfrac{1}{400}$
24. $y = x^2 + x^{-2}$ and $x$ is a positive number.
a) Write $y$ as a mixed number when:
i) $x = 1$ ii) $x = 2$ iii) $x = 3$
b) Find the value of $x$ when:
i) $y = 25.04$ ii) $y = 100.01$
👀 Show answer
ii) $2^2 + 2^{-2} = 4 + \dfrac{1}{4} = 4\dfrac{1}{4}$
iii) $3^2 + 3^{-2} = 9 + \dfrac{1}{9} = 9\dfrac{1}{9}$
b) i) $x = 5$ → $25 + \dfrac{1}{25} = 25.04$
ii) $x = 10$ → $100 + \dfrac{1}{100} = 100.01$
25. a) Write the answer to each multiplication as a power of 3:
i) $3^2 \times 3^3$
ii) $3^4 \times 3^5$
iii) $3^6 \times 3^4$
iv) $3 \times 3^5$
b) In part a you used the rule $3^a \times 3^b = 3^{a+b}$ when the indices are positive integers.
In the following multiplications, $a$ or $b$ can be negative integers.
Show that the rule still gives the correct answers:
i) $3^2 \times 3^{-1}$
ii) $3^{-2} \times 3$
iii) $3^3 \times 3^{-1}$
iv) $3^{-1} \times 3^{-1}$
v) $3^{-2} \times 3^{-1}$
c) Write two examples of your own to show that the rule works.
👀 Show answer
ii) $3^{4+5} = 3^9$
iii) $3^{6+4} = 3^{10}$
iv) $3^1 \times 3^5 = 3^{6}$
b) i) $3^2 \times 3^{-1} = 3^{1}$
ii) $3^{-2} \times 3^1 = 3^{-1}$
iii) $3^3 \times 3^{-1} = 3^2$
iv) $3^{-1} \times 3^{-1} = 3^{-2}$
v) $3^{-2} \times 3^{-1} = 3^{-3}$
c) Example 1: $3^5 \times 3^{-2} = 3^3$
Example 2: $3^{-4} \times 3^2 = 3^{-2}$
26. Write the answer to each multiplication as a power of 5:
a) $5^4 \times 5^2$
b) $5^4 \times 5^{-2}$
c) $5^{-4} \times 5^2$
d) $5^{-4} \times 5^{-2}$
👀 Show answer
b) $5^{4+(-2)} = 5^2$
c) $5^{-4+2} = 5^{-2}$
d) $5^{-6}$
12. Write the answer to each multiplication as a single power:
a) $6^{-3} \times 6^2$
b) $7^5 \times 7^{-2}$
c) $11^{-4} \times 11^{-6}$
d) $4^{-6} \times 4^2$
👀 Show answer
b) $7^3$
c) $11^{-10}$
d) $4^{-4}$
27. Find the value of $x$ in each case:
a) $2^5 \times 2^x = 2^9$
b) $3^x \times 3^{-2} = 3^4$
c) $4^x \times 4^{-3} = 4^{-5}$
d) $12^{-3} \times 12^x = 12^2$
👀 Show answer
b) $x = 6$
c) $x = -2$
d) $x = 5$
🧠 Think like a Mathematician
Question: How do you simplify divisions of powers using index laws? Does the same rule work for negative indices?
Equipment: Pencil, paper, calculator (optional)
Method:
- a) Write each of the following as a single power:
- $2^5 \div 2^3$
- $4^5 \div 4^2$
- $5^6 \div 5^5$
- $2^{10} \div 2^7$
- b) The rule for part a is: $n^a \div n^b = n^{a - b}$ when the indices $a$ and $b$ are positive integers.
Write some examples to show that this rule also works when the indices are negative integers. - c) Give your examples to a partner to check.
Follow-up Questions:
👀 show answer
- a.i:$2^5 \div 2^3 = 2^{5-3} = 2^2$
- a.ii:$4^5 \div 4^2 = 4^{5-2} = 4^3$
- a.iii:$5^6 \div 5^5 = 5^{6-5} = 5^1$
- a.iv:$2^{10} \div 2^7 = 2^{10-7} = 2^3$
- Negative indices example:$3^{-2} \div 3^{-5} = 3^{-2 - (-5)} = 3^3$
- General Rule:$n^a \div n^b = n^{a - b}$ for all integer values of $a$ and $b$
❓ EXERCISE
28. Write the answer to each division as a single power:
a) $6^2 \div 6^5$
b) $9^3 \div 9^4$
c) $15^2 \div 15^6$
d) $10^3 \div 10^8$
👀 Show answer
b) $9^{-1}$
c) $15^{-4}$
d) $10^{-5}$
29. Write the answer to each division as a single power:
a) $2^2 \div 2^{-3}$
b) $8^5 \div 8^{-2}$
c) $5^{-4} \div 5^2$
d) $12^{-3} \div 12^{-5}$
👀 Show answer
b) $8^7$
c) $5^{-6}$
d) $12^{2}$
30. Write down:
a) $8^3$ as a power of 2
b) $8^{-3}$ as a power of 2
c) $27^2$ as a power of 3
d) $27^{-2}$ as a power of 3
e) $27^2$ as a power of 9
f) $27^{-2}$ as a power of 9
👀 Show answer
b) $8^{-3} = (2^3)^{-3} = 2^{-9}$
c) $27^2 = (3^3)^2 = 3^6$
d) $27^{-2} = (3^3)^{-2} = 3^{-6}$
e) $27^2 = (9^{3/2})^2 = 9^3$
f) $27^{-2} = (9^{3/2})^{-2} = 9^{-3}$
⚠️ Be careful!
Don’t confuse negative indices with negative numbers. A negative index like $2^{-3}$ means $\dfrac{1}{2^3}$, not $-2^3$. It’s about reciprocals, not negatives!