Lesson 1 ~25 min Unit 1 · Index Laws +85 XP

Index Notation: Bases and Powers

Discover how repeated multiplication is compressed into a powerful shorthand using a base and an index (exponent).

Today's hook: A computer virus doubles every hour. After 24 hours, how many copies exist? You would need to write $2 \times 2 \times 2 \times ...$ twenty-four times. With index notation, we write $2^{24} = 16{,}777{,}216$ — one neat expression for over 16 million copies.

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Think First

warm-up

Imagine writing $3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$ on a tiny scrap of paper. Could you find a faster way to record this? What information would you need to communicate so someone else could rebuild the calculation?

Record your answer in your workbook.

The Big Idea

+5 XP

Mathematicians, like everyone, hate repetition. So we invented index notation (also called exponential notation) to compress repeated multiplication into a tiny, powerful expression.

In the expression $\,5^3$, the base is $5$ — the number being multiplied. The index (or exponent, or power) is $3$ — how many times the base is multiplied by itself. So $5^3 = 5 \times 5 \times 5 = 125$. We say "five to the power of three" or "five cubed".

$5^3 = 5 \times 5 \times 5 = 125$

Base = repeater

The big number on the bottom is the value being multiplied.

Index = how many

The small number up top counts how many copies of the base are multiplied.

Not multiplication!

$5^3 \ne 5 \times 3$. It is $5 \times 5 \times 5 = 125$, not $15$.

What You'll Master

objectives

Know

The parts of a power: base and index (exponent)
$a^n$ means $a$ multiplied by itself $n$ times
Special names: $a^2$ is "$a$ squared"; $a^3$ is "$a$ cubed"

Understand

Why index notation is more efficient than repeated multiplication
The difference between $a^n$ and $a \times n$
How exponents grow very rapidly compared to addition or multiplication

Can Do

Identify the base and index in any expression
Write repeated multiplication in index notation
Evaluate simple powers like $2^4, 3^3, 10^5$

Words You Need

vocabulary

BaseThe number being multiplied repeatedly. In $7^4$ the base is $7$.

Index / Exponent / PowerThe small number written above and to the right. It tells you how many copies of the base are multiplied.

SquaredAn index of $2$. $a^2 = a \times a$. Reads as "a squared".

CubedAn index of $3$. $a^3 = a \times a \times a$. Reads as "a cubed".

Perfect squareA number that is some integer squared: $1, 4, 9, 16, 25, 36, 49, ...$

Perfect cubeA number that is some integer cubed: $1, 8, 27, 64, 125, ...$

Spot the Trap

heads-up

Wrong: "$3^4 = 3 \times 4 = 12$." No — this confuses an index with a multiplier. $3^4 = 3 \times 3 \times 3 \times 3 = 81$.

Right: The index counts how many bases you multiply. $3^4$ means four threes multiplied together.

Wrong: "$2^5 = 10$" — treating the index as just another factor. Actually $2^5 = 32$, much bigger!

Right: Powers grow fast. Each step doubles for base $2$, triples for base $3$.

Reading Powers Aloud

+5 XP

You can read $a^n$ in several equivalent ways: "$a$ to the power of $n$", "$a$ to the $n$th", or "$a$ raised to the $n$". For specific small indices we use traditional shortcuts.

$a^2$ reads as "$a$ squared" because $a \times a$ is the area of a square with side length $a$. $a^3$ reads as "$a$ cubed" because $a \times a \times a$ is the volume of a cube with side length $a$. For $a^4, a^5, a^6, \ldots$ we just say "to the power of $4$, $5$, $6$".

$a^2$ = squared · $a^3$ = cubed · $a^n$ = $a$ to the $n$th

Squared = area

$5^2$ is the area of a $5 \times 5$ square: $25$ unit squares.

Cubed = volume

$5^3$ is the volume of a $5 \times 5 \times 5$ cube: $125$ unit cubes.

Beyond 3

There's no geometric word for $a^4$ onwards. We simply read the number.

Perfect Squares and Cubes

+5 XP

It is worth memorising the small perfect squares (up to $15^2 = 225$) and perfect cubes (up to $5^3 = 125$). These appear everywhere in algebra, geometry and surds.

Squares: $1^2=1,\;2^2=4,\;3^2=9,\;4^2=16,\;5^2=25,\;6^2=36,\;7^2=49,\;8^2=64,\;9^2=81,\;10^2=100,\;11^2=121,\;12^2=144$.

Cubes: $1^3=1,\;2^3=8,\;3^3=27,\;4^3=64,\;5^3=125,\;6^3=216,\;10^3=1000$.

Memorise these — they save you time in every test.

Watch Me Solve It · 3 examples

Watch Me Solve It · Write as a power

+15 XP per step

PROBLEM

Write $\;6 \times 6 \times 6 \times 6 \times 6\;$ in index notation, then evaluate.

1
Identify the base
The same number $6$ appears repeatedly.
Base $= 6$.
2
Count how many copies
There are five $6$s being multiplied.
Index $= 5$.
3
Write and evaluate
$6^5 = 6 \times 6 \times 6 \times 6 \times 6 = 7776$

Answer$6^5 = 7776$

Watch Me Solve It · Identify base and index

+15 XP per step

PROBLEM

For the expression $9^4$: (a) state the base, (b) state the index, (c) write it as repeated multiplication, (d) evaluate.

1
Base
The big bottom number is the base.
Base $= 9$.
2
Index
The small top number is the index.
Index $= 4$.
3
Expand and evaluate
$9^4 = 9 \times 9 \times 9 \times 9 = 81 \times 81 = 6561$

AnswerBase $9$, index $4$; $9^4 = 6561$

Watch Me Solve It · Evaluate small powers

+15 XP per step

PROBLEM

Evaluate: (a) $2^6$, (b) $10^4$, (c) $7^2$.

1
$2^6$ — double six times
$2 \to 4 \to 8 \to 16 \to 32 \to 64$
$2^6 = 64$.
2
$10^4$ — powers of ten add zeros
$10^4 = 10\,000$ (four zeros)
For base $10$, the index = number of zeros.
3
$7^2$ — a perfect square
$7 \times 7 = 49$

Answer(a) $64$ (b) $10\,000$ (c) $49$

Common Pitfalls

heads-up

Multiplying base by index

Treating $5^3$ as $5 \times 3 = 15$ is the most common error. The index is a counter, not a factor.

Fix: Always rewrite as repeated multiplication if unsure. $5^3 = 5 \times 5 \times 5$.

Confusing $3^2$ with $2^3$

Order matters! $3^2 = 9$ but $2^3 = 8$. The base and index play very different roles.

Fix: Bottom is base (repeats); top is index (counts copies). Read aloud: "three squared" vs "two cubed".

Forgetting to multiply all the way

Calculating $2^4$ as $2 \times 2 \times 2 = 8$ (only three twos) instead of $2 \times 2 \times 2 \times 2 = 16$.

Fix: Write out all the multiplications first, then compute step-by-step.

Copy Into Your Books

Index notation

$a^n = a \times a \times \ldots \times a$ ($n$ times)
$a$ = base · $n$ = index/exponent/power

Special names

$a^2$ = "$a$ squared"
$a^3$ = "$a$ cubed"
$a^n$ = "$a$ to the power of $n$"

Worth memorising

Squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$
Cubes: $1, 8, 27, 64, 125, 216$
Powers of 10: $10, 100, 1000, 10000$

Key trap

$5^3 \ne 5 \times 3$
$5^3 = 5 \times 5 \times 5 = 125$
Index counts copies, not factors

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Powers

4 problems

Four drill problems to sharpen your understanding of bases and indices.

1 Write $4 \times 4 \times 4 \times 4 \times 4 \times 4$ in index notation.
Six copies of base $4$.$4^6$
2 Evaluate $2^7$.
Double seven times: $2, 4, 8, 16, 32, 64, 128$.$2^7 = 128$
3 In $11^3$, what is the base and what is the index?
Bottom big number is the base; top small number is the index.Base $= 11$, Index $= 3$
4 Which is bigger: $3^4$ or $4^3$?
$3^4 = 81$ and $4^3 = 64$. So $3^4 > 4^3$.$3^4 = 81$ is bigger

Complete in your workbook.

Quick Check · 5 questions

Evaluate $3^5$.

+10 XP

In the expression $8^4$, identify the base and the index.

+10 XP

Write $7 \times 7 \times 7 \times 7$ in index notation.

+10 XP

Evaluate $10^5$.

+10 XP

Which of these is a perfect square?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Rewrite each expression in index notation: (a) $5 \times 5 \times 5 \times 5$, (b) $a \times a \times a \times a \times a \times a$, (c) $2 \times 2 \times 7 \times 7 \times 7$.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Evaluate: (a) $2^8$, (b) $3^4$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A bacterium splits into two every 20 minutes. Starting from $1$ bacterium, how many will there be after $4$ hours? Write your answer using index notation and as a decimal number, and explain how the index relates to time.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $3^5 = 243$.

2. A — Base $8$, index $4$.

3. B — $7^4$ (four sevens multiplied).

4. D — $10^5 = 100\,000$.

5. C — $144 = 12^2$.

Show Your Working Model Answers

Q6 (3 marks): (a) $5^4$ [1]; (b) $a^6$ [1]; (c) $2^2 \times 7^3$ [1].

Q7 (2 marks): (a) $2^8 = 256$ [1]; (b) $3^4 = 81$ [1].

Q8 (4 marks): $4$ hours $= 240$ min; $240 \div 20 = 12$ doublings [1]. After 12 doublings: $2^{12}$ bacteria [1] $= 4096$ [1]. The index $12$ counts the number of 20-minute doublings, so it represents elapsed time measured in doubling periods [1].

Stretch Challenge · +25 XP, +10 coins

The Chessboard Legend

A legend says a chess inventor asked the king to place $1$ grain of rice on square 1, $2$ on square 2, $4$ on square 3, doubling each square. How many grains are on square $20$? Write this as a power of $2$, then evaluate.

Reveal solution

Square $n$ holds $2^{n-1}$ grains. Square $20$ holds $2^{19} = 524\,288$ grains. By square $64$, the total exceeds all the rice ever grown on Earth!

Quick Review

Index notation

$a^n = a \times a \times \ldots$ ($n$ copies)

Base & index

Bottom big = base; top small = index

Names

$a^2$ squared · $a^3$ cubed

Top trap

$5^3 \ne 5 \times 3$. $5^3 = 125$.

Powers of 10

$10^n$ = 1 followed by $n$ zeros

Growth

Exponential growth is FAST — $2^{20}$ is over a million

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