Integers · Order of operations · 8.2

Powers of negatives.

Parentheses decide what gets raised. (−2)² = (−2)(−2) = 4, but −2² means −(2²) = −4 — only the 2 is squared. An even power of a negative is positive; odd stays negative.

01 / THE IDEA

See it work.

Watch the method

Powers of negatives

(−2)³ = (−2)(−2)(−2) = −8.

1 / 2
(−2)3(-2)^{3}
1

Expand the power

(-2)^3 means (-2) multiplied by itself 3 times.

(−2)(−2)(−2)(-2)(-2)(-2)
2

Count the negatives

3 negative factors → negative result.

=−8= -8
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a whole number — negatives allowed, e.g. -7.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

What’s in the parentheses

A negative inside parentheses is part of the base. Without parentheses, the exponent binds before the minus.

Why it works

(−2)² flips twice → +4; −2² is −(4).

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Evaluate (-3)^2

1

Expand the power

(-3)^2 means (-3) multiplied by itself 2 times. (-3)(-3)

2

Count the negatives

2 negative factors → positive result. = 9

EX 2medium

Evaluate -6^2

1

What is squared?

−6^2 means −(6^2); the minus is applied last — only the 6 is raised. -(6^2)

2

Raise, then negate

6^2 = 36, then apply the minus. -(36) = -36

EX 3hard

Evaluate -3^2

1

What is squared?

−3^2 means −(3^2); the minus is applied last — only the 3 is raised. -(3^2)

2

Raise, then negate

3^2 = 9, then apply the minus. -(9) = -9

04 / COMMON TRAPS

Where students slip.

Trap

Treating −2² as 4.

Fix

It is −(2²) = −4; only the 2 is squared.

Trap

Thinking (−2)³ is positive.

Fix

An odd power keeps the negative: −8.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Evaluate (-3)^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
C2

Evaluate -4^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
C3

Evaluate -6^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
C4

Evaluate (-2)^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
Apply

Apply (4)

A1

Evaluate -6^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
A2

Evaluate -4^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
A3

Evaluate -2^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
A4

Evaluate -5^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
Extension

Extension (4)

E1

Evaluate -3^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
E2

Evaluate -3^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
E3

Evaluate -5^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.
E4

Evaluate -2^2

Need a hint?
  • Parentheses? the negative is raised too.
  • No parentheses? raise the number, then negate.