📖 The Lesson
Many students do well on basic square and cube questions, but the more complicated computation questions become much harder. The good news: memorizing the common perfect squares and cubes can help a lot on the GED. On this page, memorization and pattern recognition are the whole game. The more of these you can recognize instantly, the easier every question becomes.
The big idea: these problems reward recognition, not heavy computation. If you know your perfect squares and cubes by heart and you handle negative signs carefully, most of these questions become quick.
Foundation — Squares and roots are inverses
Squaring and square-rooting undo each other
Squaring a number and taking its square root are opposite operations — each one undoes the other.
122 = 144 means √144 = 12
The same pairing works for cubes: 43 = 64 means ∛64 = 4.
These relationships should become automatic — that's what makes the GED questions fast.
Memorize the first 12 perfect squares. This is the single most useful thing you can do for this skill. A perfect square is what you get when you multiply a whole number by itself.
Perfect Squares — 12 through 122
121
224
329
4216
5225
6236
7249
8264
9281
102100
112121
122144
Memorize the first 6 perfect cubes. Cube roots show up less often than square roots, but they still appear on the GED. A perfect cube is a whole number multiplied by itself three times.
Perfect Cubes — 13 through 63
131
238
3327
4364
53125
63216
Memory hack: 4, 5, and 6 cube to numbers that end in 4, 5, and 6.
43 = 64 ·
53 = 125 ·
63 = 216
The last digit of each cube matches the number you cubed. If you blank on these three, the final digit is a built-in check: a cube of 5 has to end in 5, a cube of 6 has to end in 6, and a cube of 4 has to end in 4. That alone can rule out wrong answer choices.
The symmetry — a root just reads the chart backward
Every square and cube you memorized doubles as a
root fact. You don't learn roots separately — you read the same pairing in reverse.
| Forward (square & cube) |
Backward (square root & cube root) |
| 92 = 81 |
√81 = 9 |
| 122 = 144 |
√144 = 12 |
| 53 = 125 |
∛125 = 5 |
| 63 = 216 |
∛216 = 6 |
Same fact, read in the other direction. Knowing one direction means you already know the other.
This is the part that costs the most points. A negative sign inside parentheses is different from a negative sign outside. Slow down every time you see one.
Negative INSIDE the parentheses
(−3)2 = 9
The parentheses mean the whole thing — including the negative — gets squared.
(−3)2 = (−3) × (−3) = +9
A negative times a negative is positive.
Negative OUTSIDE the parentheses
−32 = −9
With no parentheses, you square the 3 first, then apply the negative.
−32 = −(3 × 3) = −9
The negative is not part of what gets squared.
Make it a habit: wrap a negative in parentheses. In almost every case where you are squaring or cubing a negative number, you should write it inside parentheses, like (−3)2. That is what keeps the sign attached to the number. Writing −32 without parentheses is worth understanding — it shows the negative is not part of what gets squared — but doing it that way will likely get you the wrong answer on a problem that actually means (−3)2. When in doubt, use the parentheses.
Cubes behave differently. Because three negatives multiply to a negative, a negative cube stays negative either way:
(−3)3 = (−3)(−3)(−3) = −27 and −33 = −(3×3×3) = −27.
With squares the parentheses change the sign. With cubes the answer is negative no matter where the sign sits.
The GED strategy — ask these 4 questions first
Before you compute anything, ask:
1. Is this a perfect square or a perfect cube?
2. Is the negative inside or outside the parentheses?
3. Is it asking for a square, a root, a cube, or a cube root?
This recognition process — not fast arithmetic — is the real GED skill.