Module 16 of 45Lesson 1 of 32 · Unit 11 · The Power Rules
Algebra · Unit 11
The Power Rules
Indices are shorthand for repeated multiplication — and the rules for handling them are not arbitrary. Every single one comes from counting how many copies you have. Count properly and you will never mix them up.
By the end of this unit you can
- Say what an index actually means and evaluate simple powers
- Multiply powers of the same base by adding indices
- Divide powers of the same base by subtracting indices
- Raise a power to a power by multiplying indices
- Explain why x⁰ = 1 and what a negative index means
- Handle powers that carry number coefficients
The one idea behind all of it
x⁵ means five x’s multiplied together. So x⁵ × x³ is five x’s next to three x’s — eight x’s altogether. You are not memorising a rule, you are counting:
xᵃ × xᵇ = xᵃ⁺ᵇ xᵃ ÷ xᵇ = xᵃ⁻ᵇ (xᵃ)ᵇ = xᵃᵇ
Multiply → add the indices. Divide → subtract. Power of a power → multiply.
One warning that saves more marks than any other: these rules only work when the bases are the same. x⁵ × y³ stays x⁵y³ — you cannot combine them, because you are not counting the same thing.
1
What an index means
The small number counts the copies
In 2⁵ the 2 is the base and the 5 is the index (or power). It means 2 × 2 × 2 × 2 × 2 = 32. The index counts how many copies are multiplied — it does NOT mean multiply by 5.
Worked example · Evaluate 3⁴
1
The base is 3, the index is 4
so four 3’s multiplied
2
Write it out
3 × 3 × 3 × 3
3
Work left to right
9 × 3 = 27, then 27 × 3 = 81
4
Answer
81
3⁴ is 81, not 12. Multiplying four 3’s is very different from 3 × 4 — and the gap grows fast as the index climbs.
Practice · Evaluate · 10 questions
12³
Answer8
23²
Answer9
35²
Answer25
410³
Answer1000
52⁵
Answer32
64³
Answer64
73⁴
Answer81
87²
Answer49
92¹⁰
Answer1024
1010⁶
Answer1000000
2
Multiplying powers
Same base → add the indices
x⁵ × x³ means five x’s times three x’s — eight x’s in total, so x⁸. Add the indices. This works ONLY when the bases match: x⁵ × y³ cannot be combined at all.
Every index law is just counting. Learn the counting and you never need to memorise the rule.
Worked example · Simplify a⁴ × a⁶
1
Count the a’s
four a’s and six a’s
2
Altogether
4 + 6 = 10 a’s
3
Write it
a¹⁰
Add, do not multiply. a⁴ × a⁶ is a¹⁰, not a²⁴ — you are joining two piles of a’s, not making a pile of piles.
Practice · Simplify · 10 questions
1x⁵ × x³
Answerx⁸
2a⁴ × a⁶
Answera¹⁰
3y² × y⁷
Answery⁹
4m³ × m³
Answerm⁶
5tt⁵
Answert⁶
6p⁶ × p
Answerp⁷
7x² × x³ × x⁴
Answerx⁹
8k⁸ × k²
Answerk¹⁰
9n⁵ × n⁵
Answern¹⁰
10w³ × w⁴ × w
Answerw⁸
3
Dividing powers
Same base → subtract the indices
x⁷ ÷ x³ means seven x’s on top and three underneath. Three cancel from each, leaving four: x⁴. Subtract the indices — top minus bottom, in that order.
Worked example · Simplify a⁹ ÷ a⁴
1
Nine a’s on top, four underneath
four pairs cancel
2
What is left on top
9 − 4 = 5 a’s
3
Write it
a⁵
Top minus bottom, always in that order. a⁴ ÷ a⁹ would give a⁻⁵ — a perfectly good answer, which Concept 5 explains.
Practice · Simplify · 10 questions
1x⁷x³
Answerx⁴
2a⁹a⁴
Answera⁵
3y⁶y²
Answery⁴
4m¹⁰m⁵
Answerm⁵
5t⁸t
Answert⁷
6p¹²p⁷
Answerp⁵
7k⁵k⁴
Answerk
8n²⁰n¹¹
Answern⁹
9w⁶w³
Answerw³
10z¹⁵z¹⁰
Answerz⁵
4
A power of a power
Multiply the indices
(x²)³ means x² × x² × x² — three lots of two x’s, so six x’s: x⁶. Multiply the indices. Compare carefully with Concept 2: x² × x³ adds to x⁵, but (x²)³ multiplies to x⁶.
Worked example · Simplify (a⁴)⁵
1
The bracket is repeated 5 times
a⁴ × a⁴ × a⁴ × a⁴ × a⁴
2
Each bracket brings 4 a’s
5 lots of 4 = 20
3
Write it
a²⁰
Ask yourself: is this two piles joined (add) or one pile repeated (multiply)? Brackets mean repeated — so multiply.
Practice · Simplify · 10 questions
1(x²)³
Answerx⁶
2(a⁴)⁵
Answera²⁰
3(y³)²
Answery⁶
4(m⁵)³
Answerm¹⁵
5(t²)⁷
Answert¹⁴
6(p⁶)²
Answerp¹²
7(k³)³
Answerk⁹
8(n)⁸
Answern⁸
9(w¹⁰)²
Answerw²⁰
10(z⁴)⁴
Answerz¹⁶
5
Zero and negative indices
Where the pattern leads
Follow the pattern down: 2³ = 8, 2² = 4, 2¹ = 2 — each step halves. One more step gives 2⁰ = 1, and the next gives 2⁻¹ = ½. So a negative index means one over: x⁻² = 1x².
Worked example · Why does x⁰ = 1?
1
Take any power divided by itself
x⁵ ÷ x⁵
2
Anything divided by itself is
1
3
But the index rule says subtract
x⁵⁻⁵ = x⁰
4
Both must be true, so
x⁰ = 1
x⁰ = 1 is not a strange exception someone invented. It is forced — any other value would break the division rule.
Practice · Simplify · 10 questions
15⁰
Answer1
212⁰
Answer1
3x⁰
Answer1
42⁻¹
Answer12
52⁻³
Answer18
610⁻²
Answer1100
7x⁻²
Answer1x²
8a⁻¹
Answer1a
93⁻²
Answer19
10x³x⁷
Answer1x⁴
6
Powers with coefficients
Numbers together, letters together
In 5x³ × 3x⁴, deal with the two jobs separately: multiply the numbers (5 × 3 = 15) and then the letters (x³ × x⁴ = x⁷). Answer 15x⁷. Same split for dividing.
Worked example · Simplify 12a⁷ ÷ 4a³
1
Numbers first
12 ÷ 4 = 3
2
Then letters
a⁷ ÷ a³ = a⁴
3
Put them together
3a⁴
Two separate jobs, done in order. Trying to handle the numbers and the indices in one move is where the errors live.
Practice · Simplify · 10 questions
15x³ × 3x⁴
Answer15x⁷
22a² × 6a⁵
Answer12a⁷
34y × 5y³
Answer20y⁴
412a⁷4a³
Answer3a⁴
520x⁶5x²
Answer4x⁴
6(2x³)²
Answer4x⁶
7(3a²)²
Answer9a⁴
87m⁴ × 2m
Answer14m⁵
918p⁸6p⁵
Answer3p³
10(5t²)²
Answer25t⁴
Which power rule do I use?
Look at what is happening BETWEEN the powers.
×
Multiplying powers of the same base?ADD the indices. x⁵ × x³ = x⁸.
÷
Dividing powers of the same base?SUBTRACT, top minus bottom. x⁷ ÷ x³ = x⁴.
( )
A power raised to another power?MULTIPLY the indices. (x²)³ = x⁶.
✓
Different bases, or a coefficient in front?Different bases never combine. Coefficients are a separate job — numbers with numbers, letters with letters.
The mistakes that cost the most marks
1Multiplying the index instead of counting copies
Wrong3⁴ = 12
Right3⁴ = 3 × 3 × 3 × 3 = 81
The index counts copies; it is not a multiplier. The two answers differ by a factor of nearly seven here, and far more for bigger indices.
2Multiplying indices when you should add
Wrongx⁵ × x³ = x¹⁵
Rightx⁵ × x³ = x⁸
Joining two piles of x’s ADDS them. Multiplying indices is for (x⁵)³, which is a pile repeated.
3Combining different bases
Wrongx⁵ × y³ = xy⁸
Rightx⁵ × y³ = x⁵y³ — it cannot be simplified
You can only count things that are the same. Five x’s and three y’s are different objects.
4Thinking x⁰ = 0
Wrongx⁰ = 0
Rightx⁰ = 1
Follow the halving pattern: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1. Or use x⁵ ÷ x⁵ = 1 and the subtraction rule. Both force the answer to be 1.
5Making a negative index give a negative answer
Wrong2⁻³ = −8
Right2⁻³ = 18
A negative index means RECIPROCAL, not negative. The answer of a negative power of a positive number is always positive.
Tasks to submit
Criterion B · Investigating patterns
The Zero-Power Mystery
Most students are simply told that x⁰ = 1 and asked to accept it. In this investigation you are going to discover why it has to be true — and prove that no other answer is possible.
- Write out the values of 2⁵, 2⁴, 2³, 2², 2¹ in a table. Describe exactly what happens to the value each time the index drops by 1.
- Continue your pattern for two more steps to predict 2⁰ and 2⁻¹. State what the pattern forces those values to be.
- Repeat the whole investigation with base 3 and base 10. Does the same thing happen? What is the general result?
- Now prove it a second, independent way: use xᵃ ÷ xᵃ. Write down what the answer must be by common sense, then what the subtraction rule gives, and explain why the two together force x⁰ = 1.
- Finally, classify this claim as always / sometimes / never true: “x⁰ = 1”. Investigate the single value of x that causes trouble and explain what goes wrong there.
Submit: your tables, both proofs and your written reasoning. Marked out of 8 against Criterion B. Spotting the pattern reaches the middle bands; the top band needs BOTH justifications in part 4 and a correct treatment of the awkward case in part 5.
Criterion D · Applying maths in real-life contexts
The Chain Message
A WhatsApp message asks each person to forward it to 3 new people. One person starts it. In round 1, 3 people receive it; in round 2, each of those forwards to 3 more, and so on. Botswana’s population is about 2.4 million.
- Write an expression for the number of people receiving the message in round n, and use it to complete a table for rounds 1 to 10.
- Describe how fast the numbers grow. Compare round 10 with round 5 — how many times bigger is it, and why is that number itself a power of 3?
- Find the first round in which the number of new recipients would exceed Botswana’s population. Show your working.
- Your model says the whole country is reached in a fortnight of forwarding. In reality chain messages never do this. Give three specific reasons the model overstates the spread.
- Suggest one change to the model that would make it more realistic, and explain what mathematical effect your change has on the growth.
Submit: your table, working and written explanations. Marked out of 8 against Criterion D. Parts 1–3 are the arithmetic; parts 4–5 are where the top band is decided — a model this wrong about reality is only useful if you can say WHY.
When you are ready
The mastery quiz is 10 questions drawn at random from a
bank of 25. You need 70% to pass and you have three attempts. Work through the practice questions
above first — especially any concept where you had to look at the answer more than twice.
Printable version
Download Unit 11 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit 11 as a PDF — the same lesson, laid out for printing and for working offline.