Lift the square idea into three dimensions — and watch how signs behave differently.
Stack a sugar cube on a sugar cube on a sugar cube — three across, three deep, three tall — and you have a little block of 27 sugar cubes. That number, 27, is the volume: how many unit cubes fill the box. In Lesson 6.2 you squared a side to get an area, then used the square root to walk back to the side. This lesson lifts the very same story up one dimension. Now we cube an edge to get a volume, and a brand-new operation — the cube root — walks us back to the edge. By the end you will read and evaluate symbols like 3√27, and you will discover one delightful surprise: where square roots flinch at a minus sign, cube roots do not flinch at all.
Hold the color habit steady as you read. An edge a is teal. The cube a³, the volume, is purple. The cube-root operation recovers the teal edge out of the purple volume. Keep one eye on the contrast with Lesson 6.2, where the blue square and the blue square root told the two-dimensional version of this same tale.
A square is built from one length used twice: side times side. A cube — the solid, the dice-shaped block — is built from one length used three times: edge × edge × edge. If the edge is a, that triple product is written a3 and read aloud as "a cubed" or "the cube of a." The little raised 3 is an exponent, exactly as in Lesson 6.1; here it counts three equal factors.
The word and the solid are the same idea. We call a3 "a cubed" precisely because it measures the volume of a cube whose edge is a. Picture filling that box with unit cubes — tiny 1×1×1 blocks. A cube of edge 2 holds 23 = 2×2×2 = 8 of them: two layers, four cubes per layer. A cube of edge 3 holds 33 = 3×3×3 = 27: three layers, nine per layer.
Notice how fast the volume climbs. Going from edge 2 to edge 3 is just one more unit of length, yet the count of unit cubes leaps from 8 to 27 — more than triples. That steep growth is the signature of a third power, and it is worth feeling in your hands with the widget below.
Find the volume of a cube of edge 4 cm. We cube the edge: 43 = 4×4×4. Group it: 4×4 = 16, then 16×4 = 64. So the volume is 64 cubic centimeters. The first five perfect cubes are worth knowing cold: 1, 8, 27, 64, 125 — that is 1³, 2³, 3³, 4³, 5³.
Cubing is not "times 3." A common slip is to write 5³ = 15. But 5³ means 5×5×5 = 125, not 5+5+5 and not 5×3. The exponent 3 tells you how many factors to multiply, not a number to multiply by.
Step the edge from 1 to 5 and watch the volume a³ — the number of unit cubes — build up layer by layer.
Now run the movie in reverse. Suppose someone hands you a cube and tells you it is built from 27 unit cubes, but they hide the edge from you. What is the edge? You are looking for a number whose cube is 27 — a number x that solves
The answer is 3, because 3×3×3 = 27. We write this with the cube-root symbol: 3√27 = 3, read aloud as "the cube root of twenty-seven is three." The little raised 3 tucked into the crook of the radical is the index; it announces that we are undoing a third power. The cube root does for volume exactly what the square root did for area in Lesson 6.2: it takes you from the built-up power back to the original edge.
So cubing and cube-rooting are a matched pair, an action and its undo. Cube the edge and you climb to the volume; take the cube root of the volume and you come home to the edge. 3 → 27 → 3. Whenever a volume happens to be a perfect cube — one of 1, 8, 27, 64, 125, and so on — its cube root comes out as a clean whole number.
Evaluate 3√64. Ask: which number cubed gives 64? Test from what you know — 4³ = 4×4×4 = 64. So 3√64 = 4. (A cube of edge 4 is exactly the box that holds 64 unit cubes.)
Don't confuse the index with the radicand. In 3√8, the small 3 is the index (it says "third root"), and 8 is the radicand (the number under the bar). The answer is 2, not 3: 3√8 = 2 because 2³ = 8.
Pick a perfect-cube volume, then step a guessed edge x until x³ matches it. When it locks, you have found 3√ of that volume.
Here is where the third dimension surprises us. A volume can be built from negative factors. Cube −2: that is (−2)×(−2)×(−2). Take it two factors at a time. The first two, (−2)×(−2), give +4 — two negatives make a positive. Then +4×(−2) gives −8. So (−2)³ = −8: an odd number of negative factors leaves the result negative.
Run that backward and you get a cube root that swallows a minus sign without complaint: 3√−8 = −2, because (−2)³ = −8. The sign passes straight through the cube root: a negative volume has a negative edge, a positive volume has a positive edge, and a zero has a zero. This is the headline difference from square roots, which you'll line up side by side in the next section.
And there is no ambiguity to fuss over. Every real number — positive, negative, or zero — has exactly one real cube root. There is only one number whose cube is −8 (it is −2), only one whose cube is 27 (it is 3). That is much tidier than the square case, where x² = 9 had two answers and we had to choose the principal, non-negative one.
The sign rule depends on the exponent being odd. For a square (an even power), two of the same negative cancel: (−2)² = +4, never negative. That is exactly why square roots reject negatives but cube roots accept them. Cubing keeps the sign; squaring erases it.
Toggle the base to + or −, then cube it and cube-root it back. Watch the sign survive the round trip — and notice the square root cannot make the same trip when the input is negative.
Put the two roots shoulder to shoulder and their personalities show. The square root is cautious about signs: it needs its input to be at least zero, and it always hands back something at least zero. There is no real √−9, because no real number squared is negative. The cube root has no such fear: feed it any sign and the sign simply passes out the other side.
They differ in writing, too. A square root hides a quiet index of 2 — we almost never write it, so √ alone means "second root." A cube root shows its index out loud: the small 3 in 3√ is there on purpose, to say "third root." If you ever see that little 3, you are undoing a cube, not a square.
| Square root √ | Cube root 3√ | |
|---|---|---|
| Undoes | a square (2nd power) | a cube (3rd power) |
| Hidden index | 2 (not written) | 3 (written) |
| Negative input? | not allowed | allowed |
| Sign of output | always ≥ 0 | same sign as input |
| Example | √9 = 3 | 3√−27 = −3 |
One careful reminder from Lesson 6.2 carries over. The principal square root is single and non-negative: √9 = 3, not ±3 — even though the equation x² = 9 has the two solutions x = 3 and x = −3. The cube root never makes you choose: 3√27 is just 3, the one and only real number whose cube is 27.
Feed the same number to both roots — try a perfect square or perfect cube, then flip it negative. Watch √ refuse a negative while 3√ accepts it.
Cubing uses an edge three times — a×a×a = a3 — to fill a cube with a³ unit cubes, and the cube root 3√ walks that volume back to its edge, just as the square root walked an area back to its side. The five perfect cubes 1, 8, 27, 64, 125 give clean whole-number roots. And the headline: a cube root carries the sign straight through — 3√−8 = −2 — because cubing keeps the sign, while squaring erases it and forces the square root to refuse negatives. Every real number has exactly one real cube root.
Perfect cubes and perfect squares are generous: their roots come out as tidy whole numbers. But what is √2? It is a real length — the diagonal of a unit square — yet no fraction equals it. Lesson 6.4 meets these stubborn irrational numbers and finally completes the real number line.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson serves Common Core 8.EE.A.2: students use the cube-root symbol 3√ to represent solutions of equations of the form x³ = p, and evaluate cube roots of small perfect cubes (1, 8, 27, 64, 125). It builds directly on the square-root work of 6.2 (also 8.EE.A.2) and on the exponent foundation of 6.1 (6.EE.A.1).
The #1 misconception is importing the square-root sign rule onto cube roots — assuming, as with √, that negatives are forbidden and roots must be non-negative. The antidote: make students cube a negative by hand, two factors at a time, so they see (−2)³ = −8 arises from an odd count of negative factors. Once they have built a negative volume themselves, 3√−8 = −2 stops feeling illegal — the sign simply passes through. Anchor it with the contrast: squaring erases the sign (even power), so square roots refuse negatives; cubing keeps the sign (odd power), so cube roots welcome them.