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📖 Lesson 8 ⏱ ~30 min Year 10 · Unit 1 ⚡ +50 XP

Index Laws — Multiplication and Division

Discover the shortcuts that make working with powers effortless. Why multiply when you can simply add?

Today's hook:
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Core learning
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Worksheet

Worksheet

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Warm-up
Think First
+5 XP each

Q1 · What do you already know about writing repeated multiplication in a shorter form?

Q2 · Which do you think is larger: $2^5 \times 2^3$ or $2^6$? Explain your prediction.

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From the lesson
Intentions

Learning Intentions

Know

  • Index Law 1: $a^m \times a^n = a^{m+n}$
  • Index Law 2: $a^m \div a^n = a^{m-n}$

Understand

  • Why the index laws work by expanding powers into repeated multiplication.
  • The importance of having the same base before applying index laws.

Can Do

  • Simplify expressions using the multiplication and division index laws.
  • Apply index laws to both numerical and algebraic bases.
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From the lesson
Success Criteria

Success Criteria

  • I can simplify products of powers with the same base by adding indices.
  • I can simplify quotients of powers with the same base by subtracting indices.
  • I can explain why $a^m \times a^n = a^{m+n}$ using expanded form.
  • I can identify when index laws cannot be applied (different bases).
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From the lesson
Key Terms

Key Terms

Index (exponent) — The small number that tells how many times the base is multiplied by itself. In $a^m$, $m$ is the index.
Base — The number being multiplied repeatedly. In $a^m$, $a$ is the base.
Power — The entire expression $a^m$, representing $a$ multiplied by itself $m$ times.
Index Law 1 — When multiplying powers with the same base, add the indices: $a^m \times a^n = a^{m+n}$.
Index Law 2 — When dividing powers with the same base, subtract the indices: $a^m \div a^n = a^{m-n}$.
Expanded form — Writing a power as repeated multiplication: $a^3 = a \times a \times a$.
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From the lesson
Misconceptions

Common Mistakes to Avoid

Wrong: “$2^3 \times 2^4 = 4^7$”. The base stays the same. You do not multiply the bases when multiplying powers.

Right: $2^3 \times 2^4 = 2^{3+4} = 2^7$. The base remains 2; only the indices change.

Wrong: “$3^4 \times 2^4 = 6^8$”. Index laws only apply when the bases are the same. Different bases cannot be combined this way.

Right: $3^4 \times 2^4$ cannot be simplified using index laws. It can be calculated as $81 \times 16 = 1{,}296$.

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Concept
Multiplying Powers with the Same Base
+5 XP

Why does $a^m \times a^n = a^{m+n}$? Because multiplying powers means joining together two groups of repeated multiplications.

Consider $2^3 \times 2^4$:

$2^3 = 2 \times 2 \times 2$   (three 2s)

$2^4 = 2 \times 2 \times 2 \times 2$   (four 2s)

$2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7$   (seven 2s total)

Index Law 1 — Multiplication
$a^m \times a^n = a^{m+n}$
When multiplying powers with the same base, add the indices.
Heads up

Critical condition: The bases must be identical. $x^5 \times x^3 = x^8$ works. $x^5 \times y^3$ cannot be simplified with index laws.

What to write in your book
  • When multiplying powers with the same base, add the indices: $a^m \times a^n = a^{m+n}$.
  • The base must be identical for the index law to apply.
  • Expanded form shows why the law works: repeated multiplication joins together.
Simplify $x^5 \times x^3$.
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From the lesson
Worked Example 1
Worked Example 1 — Simplifying Products
1
Given: Simplify $5^6 \times 5^3$.
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Method: The base is 5 in both terms. Add the indices: $5^6 \times 5^3 = 5^{6+3} = 5^9$.
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Answer: $\mathbf{5^9}$
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From the lesson
Worked Example 2
Worked Example 2 — Algebraic Bases
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Given: Simplify $x^4 \times x^7$.
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Method: Same base $x$. Add indices: $x^4 \times x^7 = x^{4+7} = x^{11}$.
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Answer: $\mathbf{x^{11}}$
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Concept
Dividing Powers with the Same Base
+5 XP

Why does $a^m \div a^n = a^{m-n}$? Because dividing cancels out common factors from the top and bottom.

Consider $3^5 \div 3^2$:

$\dfrac{3^5}{3^2} = \dfrac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$

Two 3s from the top cancel with the two 3s on the bottom, leaving $3 \times 3 \times 3 = 3^3$

Index Law 2 — Division
$\dfrac{a^m}{a^n} = a^{m-n}$   or   $a^m \div a^n = a^{m-n}$
When dividing powers with the same base, subtract the indices.
What to write in your book
  • When dividing powers with the same base, subtract the indices: $a^m \div a^n = a^{m-n}$.
  • Cancel common factors from the numerator and denominator to see why it works.
  • The base stays the same; only the index changes.
Simplify $\dfrac{10^9}{10^4}$.
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From the lesson
Worked Example 3
Worked Example 3 — Simplifying Quotients
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Given: Simplify $\dfrac{7^9}{7^4}$.
2
Method: Same base 7. Subtract indices: $7^{9-4} = 7^5$.
3
Answer: $\mathbf{7^5}$
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From the lesson
Worked Example 4
Worked Example 4 — Division with Algebraic Bases
1
Given: Simplify $\dfrac{y^{12}}{y^5}$.
2
Method: Same base $y$. Subtract: $y^{12-5} = y^7$.
3
Answer: $\mathbf{y^7}$
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From the lesson
Interactive

Interactive: Index Law Explorer

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From the lesson
Practice

Your Turn

Question 1: Simplify $4^3 \times 4^5$.

Question 2: Simplify $\dfrac{6^{10}}{6^7}$.

Question 3: Simplify $a^3 \times a^2 \times a^4$.

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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First answer about $2^5 \times 2^3$ vs $2^6$. Use Index Law 1 to find the exact value of $2^5 \times 2^3$. Was your prediction correct? Explain why adding the indices gives the correct answer.

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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From the lesson
MCQ 1
MCQ2 marks

Simplify $3^4 \times 3^6$.

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From the lesson
MCQ 2
MCQ2 marks

Simplify $\dfrac{5^8}{5^3}$.

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From the lesson
MCQ 3
MCQ2 marks

Which of the following CANNOT be simplified using index laws?

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From the lesson
MCQ 4
MCQ2 marks

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

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From the lesson
MCQ 5
MCQ2 marks

Simplify $\dfrac{10^7}{10^7}$.

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From the lesson
SAQ 1
Short Answer3 marks

Simplify the following expressions, leaving your answers in index form.

(a) $7^3 \times 7^8$ (1 mark)

(b) $\dfrac{m^{15}}{m^6}$ (1 mark)

(c) $2^4 \times 2^3 \times 2^5$ (1 mark)

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From the lesson
SAQ 2
Short Answer4 marks

(a) Write $5^6 \times 5^4$ as a single power and evaluate. (2 marks)

(b) Evaluate $\dfrac{4^9}{4^7}$. (2 marks)

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From the lesson
SAQ 3
Short Answer5 marks

(a) Simplify $x^7 \times x^3 \div x^4$. (2 marks)

(b) Explain why $2^3 \times 3^3$ cannot be simplified to a single power using index laws. (2 marks)

(c) A student claims that $a^5 \times a^5 = a^{25}$. Identify the error and state the correct answer. (1 mark)

R
Recap
Quick Review

Index Law 1

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

Index Law 2

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

Same base rule

Index laws only work when the bases are identical.

Multiple terms

Add or subtract all indices when several powers are combined.

Base

The number that is repeatedly multiplied.

Index (exponent)

The small number that tells how many times the base is multiplied.

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From the lesson
Real-Life Link

Real-Life Link

Index laws are the foundation of all digital technology. Every time you use a computer, your data is stored in binary — powers of 2. A kilobyte is $2^{10}$ bytes, a megabyte is $2^{20}$ bytes, and a gigabyte is $2^{30}$ bytes. Without index laws, calculating storage capacity would be impossibly tedious. In science, index laws let us work with astronomical distances and microscopic measurements using scientific notation, which you will learn in upcoming lessons.

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From the lesson
Game

Game Time!

Test your index law skills in an interactive challenge.

Play Index Laws Challenge
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From the lesson
Continue
Continue to Lesson 9: Power of a Power and Mixed Practice →
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