Lesson 3 ~25 min Unit 1 · Index Laws +85 XP

The Product Rule

When multiplying powers with the same base, you simply add the indices: $a^m \times a^n = a^{m+n}$.

Today's hook: $2^3 \times 2^4 = 2^7$. But why? Can you prove it by writing out all the multiplications? The product rule is the most important law in indices — and it follows from one beautiful observation.

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Think First

warm-up

Write $2^3 \times 2^4$ out as repeated multiplication. Count all the 2s. Can you predict what this should be as a single power of 2? Does the pattern depend on the bases being equal?

Record in your workbook.

The Big Idea

+5 XP

The product rule (or "multiplication law of indices") says: when you multiply two powers with the same base, you add the indices.

Why? Because $a^m \times a^n$ is $m$ copies of $a$ multiplied by $n$ more copies of $a$ — that's a total of $m + n$ copies of $a$, i.e. $a^{m+n}$.

$a^m \times a^n = a^{m+n}$ — same base

Same base only

$2^3 \times 3^4$ canNOT be combined — different bases.

Add the indices

$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$.

Base stays

The base does NOT change. Only the indices add.

What You'll Master

objectives

Know

$a^m \times a^n = a^{m+n}$
The product rule requires the SAME base
The base is unchanged; only indices change

Understand

Why adding indices follows from counting copies of the base
Why different bases cannot be combined this way
How to verify the rule with numerical examples

Can Do

Simplify $5^2 \times 5^7$, $a^3 \times a^4$
Identify when the rule does NOT apply
Combine more than two same-base powers

Words You Need

vocabulary

Product rule$a^m \times a^n = a^{m+n}$. Multiply same-base powers $\to$ add indices.

Same baseThe bottom number must match. $2^3 \times 2^4$ has same base $2$.

SimplifyRewrite as a single power (when possible). Example: $a^2 \times a^5 = a^7$.

ExpandWrite a power as repeated multiplication: $a^5 = a \times a \times a \times a \times a$.

CoefficientThe number in front of a variable. In $3a^2$ the coefficient is $3$.

TermA single mathematical part: $3a^2$, $-5$, $x^4$ are all terms.

Spot the Trap

heads-up

Wrong: "$2^3 \times 2^4 = 2^{12}$" — multiplying the indices instead of adding.

Right: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$. ADD the indices.

Wrong: "$2^3 \times 3^4 = 6^7$" — cannot combine different bases.

Right: $2^3 \times 3^4 = 8 \times 81 = 648$. Different bases must be evaluated separately.

Why It Works — The Proof

+5 XP

Let's verify the rule by expanding. Consider $a^3 \times a^4$:

$a^3 = a \times a \times a$ (three copies). $a^4 = a \times a \times a \times a$ (four copies). Multiplying: $a^3 \times a^4 = \underbrace{(a \cdot a \cdot a)}_{3} \times \underbrace{(a \cdot a \cdot a \cdot a)}_{4} = \underbrace{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}_{7} = a^7$.

$3 + 4 = 7$ — the indices add naturally.

Three or More Powers

+5 XP

The rule extends: $a^m \times a^n \times a^p = a^{m+n+p}$. Simply add ALL the indices that share the same base.

$5^2 \times 5^3 \times 5^4 = 5^{2+3+4} = 5^9$. If you mix bases, group same-base powers and add their indices separately: $2^3 \times 3^2 \times 2^5 = 2^{3+5} \times 3^2 = 2^8 \times 3^2$.

Add indices within each base group.

Watch Me Solve It · 3 examples

Watch Me Solve It · Simplify same-base product

+15 XP per step

PROBLEM

Simplify $7^2 \times 7^5$.

1
Check the bases
Both bases are $7$.
Product rule applies.
2
Add the indices
$2 + 5 = 7$
3
Write the result
$7^2 \times 7^5 = 7^7$

Answer$7^7$

Watch Me Solve It · Three same-base powers

+15 XP per step

PROBLEM

Simplify $a^2 \times a^4 \times a$.

1
Note the hidden index
$a = a^1$
Every variable has an implied index of $1$.
2
Add all indices
$2 + 4 + 1 = 7$
3
Result
$a^7$

Answer$a^7$

Watch Me Solve It · Different bases

+15 XP per step

PROBLEM

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

1
Group same-base powers
$(2^3 \times 2^4) \times 3^2$
2
Apply product rule to base 2
$2^{3+4} \times 3^2 = 2^7 \times 3^2$
3
Leave bases separate
$2^7 \times 3^2 = 128 \times 9 = 1152$ (evaluated)
Cannot combine $2^7$ and $3^2$ further with index laws.

Answer$2^7 \times 3^2$ (or $1152$)

Common Pitfalls

heads-up

Multiplying indices instead of adding

$a^2 \times a^3 \ne a^6$. Adding (not multiplying) gives the correct result $a^5$.

Fix: Expand to verify: $a^2 \times a^3 = (aa)(aaa) = aaaaa = a^5$.

Combining different bases

$2^3 \times 3^4 \ne 6^{12}$ and $\ne 6^7$. The rule only works for same base.

Fix: Different bases must be evaluated numerically and multiplied.

Changing the base

Some students write $2^3 \times 2^4 = 4^7$. The base must stay $2$, not double.

Fix: Base stays; only indices add. $2^3 \times 2^4 = 2^7$.

Copy Into Your Books

Product rule

$a^m \times a^n = a^{m+n}$
Same base, ADD indices
Base stays the same

Why?

$a^m \times a^n$ = $m$ + $n$ copies of $a$
Total copies $= m + n$

Three powers

$a^2 \times a^3 \times a^5 = a^{10}$
Add all indices that share a base

Mixed bases

$2^3 \times 3^4$ cannot combine
Group same-base, then apply rule

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Product rule drill

4 problems

Use the product rule to simplify each.

1 Simplify $5^4 \times 5^3$.
Same base. Add indices.$5^7$
2 Simplify $a \times a^9$.
$a = a^1$. $1 + 9 = 10$.$a^{10}$
3 Can you simplify $4^2 \times 5^2$?
Different bases — product rule fails. Evaluate: $16 \times 25$.$400$
4 Simplify $x^3 \times x^2 \times x^7$.
Add all indices: $3 + 2 + 7 = 12$.$x^{12}$

Complete in your workbook.

Quick Check · 5 questions

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

+10 XP

Simplify $x \times x^7$.

+10 XP

Which of these cannot be simplified using the product rule?

+10 XP

Evaluate $2^5 \times 2^3$.

+10 XP

Simplify $y^3 \times y^2 \times y^4$.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Simplify each: (a) $a^5 \times a^4$, (b) $7^3 \times 7$, (c) $b^2 \times b^3 \times b^4$.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. By expanding, prove that $2^3 \times 2^2 = 2^5$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A student writes "$3^4 \times 5^4 = 15^4$". Is this true or false? Justify your answer with a proof or counter-example.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $3^6$.

2. A — $x^8$.

3. D — different bases cannot combine.

4. C — $256$.

5. C — $y^9$.

Show Your Working Model Answers

Q6 (3 marks): (a) $a^9$ [1]; (b) $7^4$ [1]; (c) $b^9$ [1].

Q7 (2 marks): $2^3 \times 2^2 = (2\cdot2\cdot2)(2\cdot2) = 2\cdot2\cdot2\cdot2\cdot2 = 2^5$ [1]. Five $2$s confirms indices add: $3 + 2 = 5$ [1].

Q8 (4 marks): Actually this is TRUE because $(3 \times 5)^4 = 15^4$, which equals $3^4 \times 5^4$ by the "power of a product" rule (next lesson) [2]. Verify numerically: $3^4 \times 5^4 = 81 \times 625 = 50\,625$ [1]. $15^4 = 50\,625$ — matches [1]. (NB: this is NOT the product rule of adding indices — same INDEX, different bases means powers of a product.)

Stretch Challenge · +25 XP, +10 coins

The Hidden Index

Find the value of $n$ if $2^n \times 2^5 = 2^{12}$.

Reveal solution

By the product rule, $n + 5 = 12$, so $n = 7$.

Quick Review

Product rule

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

Same base

Required! Different bases $\to$ stuck.

Add

Indices ADD, not multiply.

$a = a^1$

Variables have an implied index of $1$.

Base unchanged

$2^3 \times 2^4 = 2^7$ (not $4^7$).

Three+ powers

Add ALL indices that share a base.

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