Mathematics • Year 9 • Unit 1 • Lesson 1
Index Notation: Bases and Powers
Build fluency with index notation: spotting the base and the index, rewriting repeated multiplication as a single power, and evaluating small powers — step by step, from a fully worked example through guided practice to independent problems.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Write $\;4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\;$ in index notation, then evaluate.
Step 1 — Spot the base.
The same number $4$ is being multiplied over and over.
Reason: the base is the number that repeats — here, $4$.
Step 2 — Count how many copies of the base there are.
There are seven $4$s being multiplied.
Reason: the index counts copies of the base — it is NOT another factor.
Step 3 — Write in index form.
$4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^7$
Reason: base $4$, index $7$, so $4^7$ — read "four to the power of seven".
Step 4 — Evaluate by doubling-up (calculator-free strategy).
$4^2 = 16$, $4^3 = 64$, $4^4 = 256$, $4^5 = 1024$, $4^6 = 4096$, $4^7 = 16{,}384$
Reason: keep multiplying by $4$ one step at a time so you don't lose track.
Answer: $4^7 = \mathbf{16{,}384}$.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Write $\;6 \times 6 \times 6 \times 6\;$ in index notation, then evaluate.
Step 1 — Spot the base: the repeated number is __________ .
Step 2 — Count the copies: there are __________ copies of the base being multiplied.
Step 3 — Write in index form:
$6 \times 6 \times 6 \times 6 = \_\_\_^{\_\_}$
Step 4 — Evaluate step by step:
$6^2 = \_\_\_\_\_$, $\;6^3 = \_\_\_\_\_$, $\;6^4 = \_\_\_\_\_$
Answer: $6^4 = \_\_\_\_\_\_$.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single skill). The middle two are standard (combine two ideas). The last two are extension (push your thinking).
Foundation — single skill
3.1 Write $\;9 \times 9 \times 9 \times 9 \times 9\;$ in index notation. 1 mark
3.2 For the expression $7^6$, state (a) the base, (b) the index. 1 mark
3.3 Evaluate $\;2^5$. 1 mark
3.4 Evaluate $\;10^4$. 1 mark
Standard — combine two ideas
3.5 Write $\;a \times a \times a \times a \times a \times a \times a \times a\;$ in index notation, and state the base. 2 marks
3.6 Write $\;3 \times 3 \times 5 \times 5 \times 5\;$ in index notation. 2 marks
Extension — push your thinking
3.7 Without using a calculator, decide which is larger: $\;3^4\;$ or $\;4^3\;$. Show both values to justify your answer. 2 marks
3.8 A perfect cube is an integer of the form $n^3$. List all perfect cubes between $1$ and $300$ inclusive. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $6 \times 6 \times 6 \times 6$)
Step 1: base is $6$.
Step 2: there are $4$ copies.
Step 3: $6 \times 6 \times 6 \times 6 = \mathbf{6^4}$.
Step 4: $6^2 = \mathbf{36}$, $6^3 = \mathbf{216}$, $6^4 = \mathbf{1296}$.
Answer: $6^4 = \mathbf{1296}$.
3.1 — Five $9$s
Base $9$, index $5$: $9 \times 9 \times 9 \times 9 \times 9 = \mathbf{9^5}$.
3.2 — $7^6$
(a) Base $= \mathbf{7}$. (b) Index $= \mathbf{6}$. (Read aloud: "seven to the power of six".)
3.3 — $2^5$
Double five times: $2 \to 4 \to 8 \to 16 \to 32$. So $2^5 = \mathbf{32}$.
3.4 — $10^4$
For base $10$, the index = number of zeros: $10^4 = \mathbf{10{,}000}$.
3.5 — Eight $a$s
Eight copies of base $a$ multiplied: $a^8$. Base is $\mathbf{a}$, index is $8$. (Common slip: writing $8a$ — that means eight $a$s added, not multiplied.) Answer: $a^8$.
3.6 — Two $3$s and three $5$s
Group same bases: two $3$s give $3^2$; three $5$s give $5^3$. So $3 \times 3 \times 5 \times 5 \times 5 = \mathbf{3^2 \times 5^3}$. (As a number, $9 \times 125 = 1125$.)
3.7 — Compare $3^4$ and $4^3$
$3^4 = 3 \times 3 \times 3 \times 3 = 81$.
$4^3 = 4 \times 4 \times 4 = 64$.
So $\mathbf{3^4 > 4^3}$ (i.e. $81 > 64$). The bigger index $4$ beats the bigger base $4$ here — index growth dominates.
3.8 — Perfect cubes between $1$ and $300$
$1^3 = 1$, $2^3 = 8$, $3^3 = 27$, $4^3 = 64$, $5^3 = 125$, $6^3 = 216$, $7^3 = 343$ (too big).
So the perfect cubes from $1$ to $300$ are: $\mathbf{1, 8, 27, 64, 125, 216}$. (Six in total.)