Mmathlab
🔥 0 🪙 0 ⚡ 0 XP ★ Lvl 1
KJ
📖 Lesson 9 ⏱ ~30 min Year 10 · Unit 1 ⚡ +50 XP

Index Laws — Power of a Power and Mixed Practice

Master the remaining index laws and learn to chain multiple laws together to simplify complex expressions in a single step.

Today's hook:
0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom
Core learning
5
From the lesson
Worksheet

Worksheet

Download or print the worksheet to work through this lesson.

Download PDF Printable HTML
Warm-up
Think First
+5 XP each

Q1 · What does $(2^3)^2$ mean in expanded form?

Q2 · Do you think $(a^2)^3$ equals $a^{2+3}$ or $a^{2 \times 3}$? Why?

6
From the lesson
Intentions

Learning Intentions

Know

  • Index Law 3: $(a^m)^n = a^{mn}$
  • Index Law 4: $(ab)^n = a^n b^n$
  • Index Law 5: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Understand

  • Why $(a^m)^n$ means multiplying the indices, not adding them.
  • How to apply multiple index laws in sequence.

Can Do

  • Simplify expressions involving power of a power, power of a product and power of a quotient.
  • Combine multiple index laws in complex simplifications.
7
From the lesson
Success Criteria

Success Criteria

  • I can simplify $(a^m)^n$ to $a^{mn}$.
  • I can expand $(ab)^n$ to $a^n b^n$ and vice versa.
  • I can simplify $\left(\dfrac{a}{b}\right)^n$ to $\dfrac{a^n}{b^n}$.
  • I can apply multiple index laws in the correct order to simplify complex expressions.
8
From the lesson
Key Terms

Key Terms

Power of a power — Raising a power to another power: $(a^m)^n$. The indices multiply.
Power of a product — Raising a product to a power: $(ab)^n = a^n b^n$. Each factor gets the power.
Power of a quotient — Raising a fraction to a power: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$. Both numerator and denominator get the power.
Index Law 3 — $(a^m)^n = a^{mn}$
Index Law 4 — $(ab)^n = a^n b^n$
Index Law 5 — $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
9
From the lesson
Misconceptions

Common Mistakes to Avoid

Wrong: “$(a^m)^n = a^{m+n}$”. This adds the indices instead of multiplying them. A power of a power means repeated multiplication of the entire power, so the indices multiply.

Right: $(a^m)^n = a^{mn}$. For example, $(2^3)^2 = 2^3 \times 2^3 = 2^{3+3} = 2^6 = 2^{3 \times 2}$.

Wrong: “$(a + b)^n = a^n + b^n$”. Index laws 4 and 5 apply to multiplication and division, NOT addition. $(2 + 3)^2 = 25$, but $2^2 + 3^2 = 13$.

Right: $(a + b)^n$ cannot be split using index laws. It must be expanded using binomial expansion (a senior topic) or calculated directly.

10
Concept
Power of a Power
+5 XP

What does $(2^3)^2$ mean? It means “take $2^3$ and square it” — so $(2^3)^2 = 2^3 \times 2^3 = 2^6$. The indices multiply.

Index Law 3 — Power of a Power
$(a^m)^n = a^{mn}$
When raising a power to another power, multiply the indices.

In expanded form:

$(2^3)^2 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 2^6$

We have 2 groups of 3 twos, which is $2 \times 3 = 6$ twos total.

What to write in your book
  • Index Law 3: $(a^m)^n = a^{mn}$ — when raising a power to another power, multiply the indices
  • Example: $(2^3)^2 = 2^{3 \times 2} = 2^6$
  • In expanded form: $(2^3)^2 = (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 2^6$
  • This is different from Index Law 1: $a^m \times a^n = a^{m+n}$
Simplify $(x^3)^4$.
11
From the lesson
Worked Example 1
Worked Example 1 — Power of a Power
1
Given: Simplify $(4^3)^5$.
2
Method: Multiply the indices: $4^{3 \times 5} = 4^{15}$.
3
Answer: $\mathbf{4^{15}}$
12
From the lesson
Worked Example 2
Worked Example 2 — Algebraic Power of a Power
1
Given: Simplify $(x^2)^7$.
2
Method: Multiply the indices: $x^{2 \times 7} = x^{14}$.
3
Answer: $\mathbf{x^{14}}$
13
Concept
Power of a Product and Quotient
+5 XP

When a product or quotient is raised to a power, every part inside the brackets receives that power.

Index Law 4 — Power of a Product
$(ab)^n = a^n b^n$
Each factor in the product gets raised to the power $n$.
Index Law 5 — Power of a Quotient
$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
Both numerator and denominator get raised to the power $n$.
Heads up

Important: These laws work for multiplication and division inside brackets, but NOT for addition or subtraction. $(a + b)^n \neq a^n + b^n$.

What to write in your book
  • Index Law 4: $(ab)^n = a^n b^n$ — each factor in the product gets raised to the power $n$
  • Index Law 5: $(a/b)^n = a^n / b^n$ — both numerator and denominator get the power $n$
  • These laws apply to multiplication and division inside brackets, NOT addition
  • Example: $(2x)^4 = 2^4 \times x^4 = 16x^4$
Simplify $(3a)^2$.
14
From the lesson
Worked Example 3
Worked Example 3 — Power of a Product
1
Given: Simplify $(2x)^4$.
2
Method: Both 2 and $x$ get the power 4: $2^4 \times x^4 = 16x^4$.
3
Answer: $\mathbf{16x^4}$
15
From the lesson
Worked Example 4
Worked Example 4 — Combined Laws
1
Given: Simplify $\dfrac{(x^3)^4 \times x^5}{x^7}$.
2
Method: Step 1: $(x^3)^4 = x^{12}$. Step 2: $x^{12} \times x^5 = x^{17}$. Step 3: $x^{17} / x^7 = x^{10}$.
3
Answer: $\mathbf{x^{10}}$
16
From the lesson
Interactive

Interactive: Mixed Index Laws Practice

17
From the lesson
Practice

Your Turn

Question 1: Simplify $(3^2)^4$.

Question 2: Simplify $(5y)^3$.

Question 3: Simplify $\dfrac{(a^2)^5 \times a^3}{a^4}$.

18
From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First answer about $(2^3)^2$. Use expanded form to show why $(2^3)^2 = 2^6$ and not $2^5$ or $2^9$. Explain the difference between adding indices (Index Law 1) and multiplying indices (Index Law 3).

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?

0
From the lesson
MCQ 1
MCQ2 marks

Simplify $(2^4)^3$.

0
From the lesson
MCQ 2
MCQ2 marks

Simplify $(3x)^4$.

0
From the lesson
MCQ 3
MCQ2 marks

Simplify $\left(\dfrac{x^4}{x^2}\right)^3$.

0
From the lesson
MCQ 4
MCQ2 marks

Which expression is equivalent to $(a^3 b^2)^4$?

0
From the lesson
MCQ 5
MCQ2 marks

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

0
From the lesson
SAQ 1
Short Answer3 marks

Simplify the following expressions.

(a) $(4^2)^3$ (1 mark)

(b) $(2a)^5$ (1 mark)

(c) $\left(\dfrac{x^6}{x^2}\right)^2$ (1 mark)

0
From the lesson
SAQ 2
Short Answer4 marks

(a) Simplify $(x^3 y^2)^4$. (2 marks)

(b) Simplify $\dfrac{(a^4)^3 \times a^2}{(a^3)^2}$. (2 marks)

0
From the lesson
SAQ 3
Short Answer5 marks

(a) A student writes $(2 + 3)^2 = 2^2 + 3^2 = 4 + 9 = 13$. Show by direct calculation that this is incorrect. (2 marks)

(b) Explain why $(a + b)^n \neq a^n + b^n$ using the meaning of brackets and repeated multiplication. (2 marks)

(c) Simplify $\dfrac{(x^2)^5 \times (x^3)^2}{(x^4)^3}$. (1 mark)

R
Recap
Quick Review

Quick Review

Law 1 — Multiplication

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

Law 2 — Division

$a^m / a^n = a^{m-n}$

Law 3 — Power of a Power

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

Law 4 — Power of a Product

$(ab)^n = a^n b^n$

Law 5 — Power of a Quotient

$(a/b)^n = a^n / b^n$

Not for Addition

$(a+b)^n \neq a^n + b^n$

0
From the lesson
Real-Life Link

Real-Life Link

Index laws power modern cryptography. When you send a secure message online, RSA encryption uses enormous powers — numbers with hundreds of digits raised to large powers. Without index laws, these calculations would be impossible. Even your phone's processor relies on breaking down complex exponentiation into manageable steps using the same laws you learned today. Scientists also use index laws when working with scales from the Planck length ($10^{-35}$ metres) to the observable universe ($10^{26}$ metres).

0
From the lesson
Game

Game Time!

Test your mixed index law skills in an interactive challenge.

Play Mixed Index Laws Challenge
0
From the lesson
Continue
Continue to Lesson 10: Zero and Negative Indices →
← Previous lesson Next lesson →