An edge of 3 builds a cube of volume 3³ = 27 small cubes. The cube root 3√27 runs the arrow backward — from the volume home to the edge.
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.
6.3.1 Cubing and the volume of a cube
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.
The edge sets the volume: 2³ = 8 unit cubes on the left, 3³ = 27 on the right. Each step up the edge adds a whole new slab of little cubes.
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.
Worked example
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.
Watch out
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.
🎮 Try itGrow the cube, count the unit cubes
Step the edge from 1 to 5 and watch the volume a³ — the number of unit cubes — build up layer by layer.
Edge a3
6.3.2 Taking a cube root: from volume back to edge
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
Solving x³ = 27 means undoing the cube. The cube-root symbol 3√ names that undoing exactly.
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.
Worked example
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.)
Watch out
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.
🎮 Try itFind the hidden edge
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.
Volume
Guess edge x2
6.3.3 How cube roots handle signs
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.
Cubing keeps the sign of the base; cube-rooting hands that same sign back. 3√−8 = −2 and 3√8 = 2.
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.
Watch out
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.
🎮 Try itThe sign-through-the-cube-root machine
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.
Base
Size2
6.3.4 Square roots versus cube roots
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.
Same shaped symbol, different jobs. The square root carries a silent index 2; the cube root wears a visible index 3.
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 equationx² = 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.
🎮 Try it√ versus ∛ comparator
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.
Sign
Number
★ The big ideas, in one breath
Cubing uses an edge three times — a×a×a = a3 — to fill a cube with a³ unit cubes, and the cube root3√ 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.
Coming up next — 6.4
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.
✎ Exercises 6.3
Work each one out first, then open the answer to check your thinking.
Write 2×2×2 as a power, then find its value.
Show answer
Three equal factors of 2 make 23, read "two cubed." Its value is 2×2×2 = 8.
A cube has edge 3 cm. What is its volume?
Show answer
Cube the edge: 33 = 3×3×3 = 27. The volume is 27 cubic centimeters.
Evaluate 3√27.
Show answer
Ask which number cubed gives 27. Since 3³ = 27, we get 3√27 = 3.
Evaluate 3√1 and 3√0.
Show answer
1³ = 1, so 3√1 = 1. And 0³ = 0, so 3√0 = 0.
Solve x³ = 64 for the edge x.
Show answer
We want the number whose cube is 64. Since 4³ = 4×4×4 = 64, the edge is x = 4, that is 3√64 = 4.
Compute (−2)³. Show the sign step by step.
Show answer
Two factors first: (−2)×(−2) = +4. Then +4×(−2) = −8. An odd number of negative factors leaves a negative result: (−2)³ = −8.
Evaluate 3√−27.
Show answer
The sign passes through a cube root. We need the number whose cube is −27. Since (−3)³ = −27, we get 3√−27 = −3.
True or false: √−9 is a real number. Explain.
Show answer
False. No real number squared is negative, since a square of any real is ≥ 0. So √−9 has no real value. (Contrast: 3√−27 = −3 is perfectly real, because a cube can be negative.)
A cube has volume 125 m³. Find its edge, then say how many edges along it differs from a cube of volume 64 m³.
Show answer
3√125 = 5 because 5³ = 125, so the edge is 5 m. A volume of 64 m³ has edge 3√64 = 4 m. The edges differ by 5 − 4 = 1 meter — even though the volumes differ by 61 cubic meters.
Evaluate and compare: −4³ versus (−4)³. Are they equal? Then give 3√−64.
Show answer
For a cube (an odd power) these two do land in the same place, but read the steps carefully. −4³ means "the negative of 4³," that is −(64) = −64. And (−4)³ = (−4)×(−4)×(−4) = −64 too. So here they are equal, both −64. (Careful: for an even power they would differ — −4² = −16 but (−4)² = +16.) Finally, 3√−64 = −4, since (−4)³ = −64.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.