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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