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Lesson 6 ~25 min Unit 1 · Index Laws +85 XP

Combining Index Laws

When two or three rules collide in one expression: simplify inside brackets first, multiply next, divide last.

Today's hook: An expression has three index laws in it. Apply them in the wrong order and you get a different answer. Strategy matters!
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Think First
warm-up

Simplify $2^3 \times 2^5 \div 2^4$. Which rule do you use first, the product rule or the quotient rule? Does the order change your answer?

Record in your workbook.
1
The Big Idea
+5 XP

Real expressions usually need more than one index law. The trick is choosing a sensible order so you simplify, not complicate.

Strategy: deal with brackets first (power-of-a-power), then multiply (product rule), then divide (quotient rule). Keep the same base.

Brackets $\to$ multiply $\to$ divide
Product
$a^m \times a^n = a^{m+n}$. Same base, add indices.
Quotient
$a^m \div a^n = a^{m-n}$. Same base, subtract.
Power-of-power
$(a^m)^n = a^{mn}$. Multiply indices.
2
What You'll Master
objectives

Know

  • Product rule: $a^m \times a^n = a^{m+n}$
  • Quotient rule: $a^m \div a^n = a^{m-n}$
  • Power of a power: $(a^m)^n = a^{mn}$

Understand

  • Why brackets are simplified first
  • Why order changes intermediate steps but not the final answer (if rules used correctly)
  • How to check by evaluating numerically

Can Do

  • Simplify $2^3 \times 2^5 \div 2^4$
  • Simplify $(3^2)^3 \times 3^4$
  • Verify answers by direct calculation
3
Words You Need
vocabulary
CombineApply more than one index law in sequence.
Order of operationsBrackets before multiplication, multiplication before division.
Same baseAll terms must share the same base before you add or subtract indices.
StrategyA planned order for applying rules.
VerifyCheck by evaluating the expression directly.
SequenceThe step-by-step order rules are applied.
4
Spot the Trap
heads-up

Wrong: "$(3^2)^3 \times 3^4 = 3^2 \times 3^{3+4}$", applying the product rule before resolving the bracket.

Right: Resolve bracket first: $(3^2)^3 = 3^6$, then $3^6 \times 3^4 = 3^{10}$.

Wrong: "$2^3 \times 2^5 \div 2^4 = 2^{3 \times 5 - 4} = 2^{11}$", multiplying indices instead of adding.

Right: $2^{3+5-4} = 2^4 = 16$. Product = add; quotient = subtract.

5
A Strategy You Can Trust
+5 XP

Whenever an expression mixes rules, follow this checklist:

1. Brackets apply $(a^m)^n = a^{mn}$ first.
2. Multiply combine factors with the same base: $a^m \times a^n = a^{m+n}$.
3. Divide finish with $a^m \div a^n = a^{m-n}$.
4. Check if the numbers are small, evaluate directly.

B $\to$ M $\to$ D $\to$ Check
6
Always Verify
+5 XP

When the base is small (like $2$ or $3$), you can re-do the problem by direct calculation as a check.

For $2^3 \times 2^5 \div 2^4$: by index laws we get $2^4 = 16$. Verify directly: $8 \times 32 \div 16 = 256 \div 16 = 16$. The two methods agree, we know we got it right.

$2^3 \times 2^5 \div 2^4 = 2^4 = 16$

Watch Me Solve It · 3 examples

Watch Me Solve It · Product then quotient
+15 XP per step
Q1
PROBLEM
Simplify $2^3 \times 2^5 \div 2^4$.
  1. 1
    Product rule first
    $2^3 \times 2^5 = 2^{3+5} = 2^8$
    Same base, indices add.
  2. 2
    Quotient rule
    $2^8 \div 2^4 = 2^{8-4} = 2^4$
  3. 3
    Evaluate & check
    $2^4 = 16$
    Verify: $8 \times 32 \div 16 = 16$ ✓
Answer$2^4 = 16$
Watch Me Solve It · Bracket then product
+15 XP per step
Q2
PROBLEM
Simplify $(3^2)^3 \times 3^4$.
  1. 1
    Resolve the bracket
    $(3^2)^3 = 3^{2 \times 3} = 3^6$
    Power-of-a-power: multiply indices.
  2. 2
    Apply product rule
    $3^6 \times 3^4 = 3^{6+4}$
  3. 3
    Final index
    $3^{10}$
Answer$3^{10}$
Watch Me Solve It · Bracket then quotient
+15 XP per step
Q3
PROBLEM
Simplify $\dfrac{(5^3)^2}{5^4}$.
  1. 1
    Resolve the bracket
    $(5^3)^2 = 5^{3 \times 2} = 5^6$
  2. 2
    Apply quotient rule
    $\dfrac{5^6}{5^4} = 5^{6-4} = 5^2$
  3. 3
    Evaluate
    $5^2 = 25$
    Verify: $\dfrac{15625}{625} = 25$ ✓
Answer$5^2 = 25$

Common Pitfalls

8
Common Pitfalls
heads-up
Skipping the bracket
$(3^2)^3 \times 3^4 \ne 3^2 \times 3^7$. You can't combine the inner $3^2$ with $3^4$ until the bracket is resolved.
Fix: Always resolve $(a^m)^n$ first, brackets before everything else.
Mixing add and multiply
Product rule ADDS indices; power-of-a-power MULTIPLIES them. Easy to confuse under pressure.
Fix: Brackets $\to$ multiply indices. Times sign with same base $\to$ add indices.
Different bases
$2^3 \times 3^4 \ne 6^7$. You can only add indices when the bases match.
Fix: Check bases are the same before combining.
Copy Into Your Books

Strategy

  • 1. Brackets first
  • 2. Multiply (product rule)
  • 3. Divide (quotient rule)
  • 4. Check with numbers

Rules summary

  • $a^m \times a^n = a^{m+n}$
  • $a^m \div a^n = a^{m-n}$
  • $(a^m)^n = a^{mn}$

Example

  • $(3^2)^3 \times 3^4$
  • $= 3^6 \times 3^4$
  • $= 3^{10}$

Always check

  • Same base?
  • Bracket resolved first?
  • Add or multiply?

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Combining laws
4 problems

Four drills to lock in the strategy.

  1. 1 Simplify $2^4 \times 2^2 \div 2^3$.

    Add then subtract.$2^3 = 8$

  2. 2 Simplify $(a^3)^2 \times a^5$.

    Bracket: $a^6$, then add 5.$a^{11}$

  3. 3 Simplify $\dfrac{(2^4)^2}{2^5}$.

    $2^8 \div 2^5$.$2^3 = 8$

  4. 4 Simplify $x^2 \times (x^3)^2 \div x^4$.

    $x^2 \times x^6 \div x^4 = x^{2+6-4}$.$x^4$
Complete in your workbook.

Quick Check · 5 questions

1
Simplify $2^3 \times 2^5 \div 2^4$.
+10 XP
2
Simplify $(3^2)^3 \times 3^4$.
+10 XP
3
Simplify $\dfrac{(5^3)^2}{5^4}$.
+10 XP
4
Simplify $x^2 \times x^3 \div x^4$.
+10 XP
5
Simplify $(2^3)^2 \times 2^2 \div 2^5$.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
ApplyMedium3 MARKS

Q6. Simplify each: (a) $4^3 \times 4^2 \div 4^4$, (b) $(a^2)^4 \div a^3$, (c) $(2^3)^2 \times 2^4$.

Answer in your workbook.
UnderstandEasy2 MARKS

Q7. A student writes $(a^2)^3 \times a^4 = a^{2+3+4} = a^9$. Identify the error and show the correct working leading to $a^{10}$.

Answer in your workbook.
ReasonHard4 MARKS

Q8. Simplify $\dfrac{(2^4)^2 \times 2^3}{2^6}$ and then verify your answer by direct calculation.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. C$2^4 = 16$.

2. D$3^{10}$.

3. A$25$.

4. B$x$.

5. C$8$.

Show Your Working Model Answers

Q6 (3 marks): (a) $4^{3+2-4} = 4^1 = 4$ [1]; (b) $a^{8-3} = a^5$ [1]; (c) $2^6 \times 2^4 = 2^{10}$ [1].

Q7 (2 marks): The student added all three indices, but the bracket $(a^2)^3$ requires MULTIPLYING: $(a^2)^3 = a^6$ [1]. Then product rule: $a^6 \times a^4 = a^{10}$ [1].

Q8 (4 marks): $(2^4)^2 = 2^8$ [1]. Numerator: $2^8 \times 2^3 = 2^{11}$ [1]. Divide: $2^{11} \div 2^6 = 2^5 = 32$ [1]. Verify: $\dfrac{256 \times 8}{64} = \dfrac{2048}{64} = 32$ ✓ [1].

Stretch Challenge · +25 XP, +10 coins

Mixed Bag

Simplify $\dfrac{(a^3)^4 \times a^2}{(a^5)^2}$ giving your answer as a single power of $a$.

Reveal solution

Top: $a^{12} \times a^2 = a^{14}$. Bottom: $a^{10}$. So $a^{14-10} = a^4$.

R
Quick Review

Brackets first

$(a^m)^n = a^{mn}$

Product

$a^m \times a^n = a^{m+n}$

Quotient

$a^m \div a^n = a^{m-n}$

Strategy

B $\to$ M $\to$ D $\to$ Check

Same base

Required for product/quotient

Verify

Re-do with numbers when small

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