Lesson 5 ~25 min Unit 1 · Index Laws +85 XP

The Power of a Power Rule

When you raise a power to another power, multiply the indices: $(a^m)^n = a^{mn}$.

Today's hook: $(2^3)^4$: do you multiply $3 \times 4$ or add $3 + 4$? One gives $2^{12}$ ($= 4096$), the other gives $2^7$ ($= 128$). Huge difference!

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Think First

warm-up

Expand $(2^3)^4$ as $2^3 \times 2^3 \times 2^3 \times 2^3$ and use the product rule to simplify. What single power of $2$ do you get? Do you spot a shortcut?

Record in your workbook.

The Big Idea

+5 XP

The power of a power rule says: when raising a power to another power, MULTIPLY the indices.

$(a^m)^n$ means "$a^m$ multiplied by itself $n$ times". That gives $n$ copies of $m$ factors of $a$ — a total of $mn$ factors of $a$.

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

Multiply, don't add

$(2^3)^4 = 2^{3 \times 4} = 2^{12}$, NOT $2^7$.

Power of a product

$(ab)^n = a^n b^n$ — every factor gets the index.

Power of a quotient

$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$, $b \ne 0$.

What You'll Master

objectives

Know

$(a^m)^n = a^{mn}$
$(ab)^n = a^n b^n$
$(a/b)^n = a^n / b^n$ for $b \ne 0$

Understand

Why multiplying indices follows from counting copies of copies
The difference between this rule and the product rule (which ADDS)
How brackets signal which rule applies

Can Do

Simplify $(3^4)^2, (a^5)^3$
Apply $(ab)^n$ to expressions like $(2x)^3$
Distinguish $(a^m)^n$ from $a^m \times a^n$

Words You Need

vocabulary

Power of a power$(a^m)^n = a^{mn}$

Power of a product$(ab)^n = a^n b^n$

Power of a quotient$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Outer indexThe index outside the bracket, $n$ in $(a^m)^n$.

Inner indexThe index inside the bracket, $m$ in $(a^m)^n$.

DistributeApply a power to every factor inside brackets.

Spot the Trap

heads-up

Wrong: "$(2^3)^4 = 2^7$" — adding instead of multiplying.

Right: $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Wrong: "$(2x)^3 = 2x^3$" — forgetting to raise the $2$.

Right: $(2x)^3 = 2^3 x^3 = 8x^3$. EVERY factor gets the power.

Power of a Product

+5 XP

When a product is inside brackets, the outer index applies to every factor inside.

$(ab)^n = (ab)(ab)\cdots(ab)$ ($n$ times) $= (a \cdot a \cdots a)(b \cdot b \cdots b) = a^n b^n$. Every factor inside the brackets gets the power.

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

Power of a Quotient

+5 XP

The same idea works for division: raise both top and bottom by the index.

$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ for $b \ne 0$. Example: $\left(\dfrac{2}{5}\right)^3 = \dfrac{2^3}{5^3} = \dfrac{8}{125}$.

$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Watch Me Solve It · 3 examples

Watch Me Solve It · Power of a power

+15 XP per step

PROBLEM

Simplify $(7^3)^4$.

1
Apply rule
$(a^m)^n = a^{mn}$. So $(7^3)^4 = 7^{3 \times 4}$
2
Multiply indices
$3 \times 4 = 12$
3
Result
$7^{12}$

Answer$7^{12}$

Watch Me Solve It · Power of a product

+15 XP per step

PROBLEM

Simplify $(2x)^4$.

1
Apply to each factor
$(2x)^4 = 2^4 \cdot x^4$
Every factor gets the index.
2
Evaluate the number
$2^4 = 16$
3
Combine
$16x^4$

Answer$16x^4$

Watch Me Solve It · Power of a quotient

+15 XP per step

PROBLEM

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

1
Apply rule
$\left(\dfrac{3}{a}\right)^2 = \dfrac{3^2}{a^2}$
2
Evaluate top
$\dfrac{9}{a^2}$
3
Check restriction
$a \ne 0$
Otherwise the original is undefined.

Answer$\dfrac{9}{a^2}$

Common Pitfalls

heads-up

Adding instead of multiplying

$(a^3)^4 \ne a^7$. The power-of-a-power rule MULTIPLIES indices.

Fix: Look for brackets! Power inside brackets, power outside = multiply.

Forgetting all factors

$(2x)^3 \ne 2x^3$. The $2$ also gets raised: $(2x)^3 = 8x^3$.

Fix: EVERY factor inside the bracket gets the outer index.

Mixing rules

$a^3 \times a^4 \ne a^{12}$. Product rule ADDS; only power-of-a-power multiplies.

Fix: Adding indices = multiplication; multiplying indices = power-of-power.

Copy Into Your Books

Power of a power

$(a^m)^n = a^{mn}$
MULTIPLY indices
Different from product rule (ADD)

Power of a product

$(ab)^n = a^n b^n$
$(2x)^3 = 8x^3$
Every factor gets the index

Power of a quotient

$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$
$b \ne 0$

Compare

Product: $a^m \times a^n = a^{m+n}$
Power-of-power: $(a^m)^n = a^{mn}$
Add vs multiply!

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Power-of-power drill

4 problems

Four drills to lock in this rule.

1 Simplify $(a^5)^2$.
Multiply indices.$a^{10}$
2 Simplify $(3y)^3$.
Each factor to the 3.$27y^3$
3 Simplify $(2^4)^3$.
$4 \times 3 = 12$.$2^{12}$
4 Simplify $\left(\dfrac{x}{2}\right)^4$.
Both top and bottom raised.$\dfrac{x^4}{16}$

Complete in your workbook.

Quick Check · 5 questions

Simplify $(5^2)^3$.

+10 XP

Expand $(3x)^4$.

+10 XP

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

+10 XP

Which formula is the power-of-a-power rule?

+10 XP

Simplify $(2a^3)^4$.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Simplify each: (a) $(a^4)^5$, (b) $(2y)^3$, (c) $\left(\dfrac{x}{3}\right)^2$.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. By expanding, prove that $(a^2)^3 = a^6$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. Simplify $(2x^2 y)^3$ fully and explain in words why the coefficient becomes $8$.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $5^6$.

2. D — $81x^4$.

3. A — $\dfrac{a^2}{16}$.

4. C — $(a^m)^n = a^{mn}$.

5. B — $16a^{12}$.

Show Your Working Model Answers

Q6 (3 marks): (a) $a^{20}$ [1]; (b) $8y^3$ [1]; (c) $\dfrac{x^2}{9}$ [1].

Q7 (2 marks): $(a^2)^3 = a^2 \times a^2 \times a^2 = a^{2+2+2} = a^6$ [1]. Matches $2 \times 3 = 6$ [1].

Q8 (4 marks): $(2x^2 y)^3 = 2^3 (x^2)^3 y^3$ [1] $= 8 x^6 y^3$ [2]. The $2$ is raised to the power $3$ giving $8$, because the outer index applies to every factor inside the brackets, including numerical coefficients [1].

Stretch Challenge · +25 XP, +10 coins

Triple Power

Find the value of $n$ if $(2^n)^3 = 2^{18}$.

Reveal solution

$3n = 18$, so $n = 6$.

Quick Review

Power of power

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

Power of product

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

Power of quotient

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

MULTIPLY indices

Different from product rule (ADD)

Every factor

All factors get the outer index

$(2x)^3 = 8x^3$

Don't forget the coefficient!

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