Lesson 2 ~25 min Unit 1 · Index Laws +85 XP

Evaluating Powers

Sharpen your skills evaluating positive, negative, and fractional bases — and learn the crucial difference between $-2^4$ and $(-2)^4$.

Today's hook: Is $-2^4$ the same as $(-2)^4$? One equals $-16$, the other equals $+16$. Tiny brackets, huge difference. Can you work out which is which — before someone marks your test?

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Think First

warm-up

What is the value of $(-3)^2$? And what is $-3^2$? They look almost the same on the page — but they give different answers. Can you predict which is positive and which is negative, and why?

Record in your workbook.

The Big Idea

+5 XP

Evaluating a power is not just multiplication — it is reading the expression carefully. Brackets, signs and indices interact in ways that catch most students out.

The brackets tell you what the base actually is. In $(-2)^4$, the base is $-2$ — the negative is inside the bracket, so it's part of the base. In $-2^4$, the base is just $2$; the negative sits outside and applies after the power is calculated. So $(-2)^4 = 16$ but $-2^4 = -16$.

$(-2)^4 = 16$ vs $-2^4 = -16$

Brackets first

If the negative is inside brackets, it is part of the base.

Even = positive

$(-)\!^{\text{even}}$ gives a positive answer.

Odd = negative

$(-)\!^{\text{odd}}$ stays negative.

What You'll Master

objectives

Know

$a^1 = a$ and (looking ahead) $a^0 = 1$ for $a \ne 0$
Negative base raised to even index is positive; odd index keeps sign
$(a/b)^n = a^n / b^n$

Understand

The difference between $-a^n$ and $(-a)^n$
Why brackets are essential for negative bases
Order of operations including powers

Can Do

Evaluate $(-3)^4, -3^4$, $(1/2)^3$, $(-2)^5$ accurately
Apply BIDMAS with powers in mixed expressions
Predict signs of powers without full calculation

Words You Need

vocabulary

Even indexAn index that is $2, 4, 6, ...$. A negative base raised to an even index gives a positive answer.

Odd indexAn index that is $1, 3, 5, ...$. A negative base raised to an odd index gives a negative answer.

Brackets matter$(-a)^n$ raises the negative; $-a^n$ raises only $a$, then applies the negative.

Power of a fraction$(a/b)^n = a^n / b^n$. Raise top and bottom separately.

$a^1$Any number to the power of $1$ equals itself.

$1^n$$1$ raised to any power is always $1$, since $1 \times 1 \times \ldots = 1$.

Spot the Trap

heads-up

Wrong: "$-3^2 = 9$" — treating the negative as part of the base.

Right: $-3^2 = -(3^2) = -9$. The negative is OUTSIDE; the base is just $3$.

Wrong: "$(\tfrac{1}{2})^3 = \tfrac{1}{6}$" — multiplying bottom by index.

Right: $(\tfrac{1}{2})^3 = \tfrac{1^3}{2^3} = \tfrac{1}{8}$. Raise top and bottom by the index.

Negative Bases — The Sign Rule

+5 XP

When you multiply two negatives, the result is positive. So $(-2) \times (-2) = +4$. Multiplying another $(-2)$ gives a negative again: $(-2)^3 = -8$.

$(-2)^2 = (-2)(-2) = 4$. $(-2)^3 = (-2)(-2)(-2) = -8$. $(-2)^4 = 16$. $(-2)^5 = -32$. The pattern: even index $\to$ positive; odd index $\to$ negative.

$(-a)^n = a^n$ when $n$ is even; $-a^n$ when $n$ is odd.

Count negatives

Each $(-)$ flips the sign. Even count $\to$ positive.

Add brackets

If the base is negative, ALWAYS use brackets to avoid confusion.

No brackets?

$-a^n$ means $-(a^n)$. Always negative if $n \ge 1$.

Powers of Fractions and Order of Operations

+5 XP

For a fraction, raise both the numerator and denominator. For mixed expressions, follow BIDMAS — powers come before multiply/divide and add/subtract.

$\left(\dfrac{2}{3}\right)^3 = \dfrac{2^3}{3^3} = \dfrac{8}{27}$. In mixed expressions like $5 + 3 \times 2^3$, calculate the power first: $2^3 = 8$, then $3 \times 8 = 24$, then $5 + 24 = 29$.

BIDMAS: Indices come BEFORE multiply/divide.

Watch Me Solve It · 3 examples

Watch Me Solve It · Bracket trap

+15 XP per step

PROBLEM

Evaluate (a) $(-3)^4$ and (b) $-3^4$. Why do they differ?

1
(a) $(-3)^4$ — negative is inside
$(-3) \times (-3) \times (-3) \times (-3)$
$= 9 \times 9 = 81$ (even index $\to$ positive).
2
(b) $-3^4$ — negative is outside
$-(3^4) = -(81) = -81$
3
Compare
$(-3)^4 = 81$ but $-3^4 = -81$. Brackets change the base.

Answer(a) $81$ (b) $-81$

Watch Me Solve It · Fraction power

+15 XP per step

PROBLEM

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

1
Raise both top and bottom
$\left(\dfrac{3}{4}\right)^2 = \dfrac{3^2}{4^2}$
2
Evaluate each
$3^2 = 9$, $4^2 = 16$
3
Combine
$= \dfrac{9}{16}$

Answer$\dfrac{9}{16}$

Watch Me Solve It · Order of operations

+15 XP per step

PROBLEM

Evaluate $2 + 3 \times 4^2$.

1
Index first (BIDMAS)
$4^2 = 16$
2
Multiplication next
$3 \times 16 = 48$
3
Addition last
$2 + 48 = 50$

Answer$50$

Common Pitfalls

heads-up

$-3^2$ vs $(-3)^2$

Without brackets, the negative is NOT raised. $-3^2 = -9$, but $(-3)^2 = 9$.

Fix: Identify the base FIRST. If unsure, rewrite as $-(3^2)$ vs $(-3)(-3)$.

Fraction power errors

Multiplying denominator by index: $(\tfrac{1}{2})^4 \ne \tfrac{1}{8}$. The denominator is raised too.

Fix: $(\tfrac{a}{b})^n = \tfrac{a^n}{b^n}$. Top to $n$, bottom to $n$.

Ignoring BIDMAS

Calculating $5 + 2^3$ left-to-right as $7^3 = 343$ is wrong. Indices come first.

Fix: Always do indices before $+, -, \times, \div$.

Copy Into Your Books

Negative bases

$(-a)^{\text{even}}$ = positive
$(-a)^{\text{odd}}$ = negative
$-a^n = -(a^n)$ always

Fraction powers

$(\tfrac{a}{b})^n = \tfrac{a^n}{b^n}$
$(\tfrac{1}{2})^3 = \tfrac{1}{8}$
$(\tfrac{2}{3})^2 = \tfrac{4}{9}$

Specials

$a^1 = a$
$1^n = 1$
$(-1)^{\text{even}} = 1$; $(-1)^{\text{odd}} = -1$

BIDMAS reminder

Brackets, Indices, Divide/Multiply, Add/Subtract
Indices come BEFORE multiplication

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Sign & bracket drill

4 problems

Four problems to lock in sign rules and bracket handling.

1 Evaluate $(-5)^2$.
Even index, base is $-5$.$25$
2 Evaluate $-5^2$.
Base is just $5$; negative applies after.$-25$
3 Evaluate $\left(\dfrac{1}{3}\right)^3$.
Cube both top and bottom.$\dfrac{1}{27}$
4 Evaluate $10 - 2^3$.
Index first: $2^3 = 8$. Then $10 - 8$.$2$

Complete in your workbook.

Quick Check · 5 questions

Evaluate $(-2)^5$.

+10 XP

Evaluate $-4^2$.

+10 XP

Evaluate $\left(\dfrac{2}{5}\right)^2$.

+10 XP

Evaluate $4 + 2 \times 3^2$.

+10 XP

Evaluate $(-1)^{100}$.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Evaluate each, showing one line of working: (a) $(-2)^6$, (b) $-2^6$, (c) $\left(\tfrac{3}{5}\right)^2$.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Use BIDMAS to evaluate $20 - 3 \times 2^2$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. Explain, with examples, why $(-a)^n$ and $-a^n$ give the same answer when $n$ is odd, but different answers when $n$ is even. Use $a = 3$ with $n = 2, 3, 4$ to illustrate.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $(-2)^5 = -32$.

2. D — $-4^2 = -16$.

3. A — $(\tfrac{2}{5})^2 = \tfrac{4}{25}$.

4. C — $22$.

5. A — $(-1)^{100} = 1$.

Show Your Working Model Answers

Q6 (3 marks): (a) $(-2)^6 = 64$ [1]; (b) $-2^6 = -64$ [1]; (c) $\tfrac{9}{25}$ [1].

Q7 (2 marks): $2^2 = 4$ [1]. $20 - 3 \times 4 = 20 - 12 = 8$ [1].

Q8 (4 marks): $n=2$: $(-3)^2 = 9$, $-3^2 = -9$ — different [1]. $n=3$: $(-3)^3 = -27$, $-3^3 = -27$ — same [1]. $n=4$: $(-3)^4 = 81$, $-3^4 = -81$ — different [1]. Odd indices preserve the negative sign so $(-a)^n = -a^n$; even indices flip negatives to positive, so they differ [1].

Stretch Challenge · +25 XP, +10 coins

The Mystery Power

If $(-x)^n = -x^n$ for a positive number $x$ and a whole-number index $n$, what can you say about $n$? Justify your answer.

Reveal solution

$(-x)^n$ equals $x^n$ when $n$ is even (a positive value) and $-x^n$ when $n$ is odd. For $(-x)^n = -x^n$ we need $n$ odd.

Quick Review

Sign rule

$(-)^{\text{even}} = +$, $(-)^{\text{odd}} = -$

Bracket matters

$(-3)^2 = 9$, $-3^2 = -9$

Fraction power

$(\tfrac{a}{b})^n = \tfrac{a^n}{b^n}$

$1^n$ & $0^n$

$1^n = 1$; $0^n = 0$ for $n \ge 1$

BIDMAS

Indices BEFORE multiply/divide

Memory tip

"No brackets = no negative in the base"

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