Before this page: Know how to read a power and why a root symbol gives one value.
Finding the edge of a cube
Imagine packing 27 small cubes into one larger cube. How many fit along each edge? Three fit across, three fit back, and there are three layers: 3 × 3 × 3 = 27.
3 × 3 × 3 = 27
Each layer has 3 × 3 = 9 cubes.
Three layers make 27 cubes.
The cube root works backwards from that volume to the edge length. We write 27 = 3. The small 3 tells us to look for three equal factors, rather than the two used for a square root.
A cube root doesn't mean dividing by 3. It means finding a number that, multiplied by itself and then by itself again, gives the original number.
Cube roots of negative numbers
A physical cube can't have a negative volume, but a number can have a negative cube. Start with −3 and multiply:
(−3) × (−3) = 9
9 × (−3) = −27
The first two negative factors give a positive product. The third makes it negative again. So −27 = −3.
Unlike squaring, cubing keeps the sign. A positive number has a positive cube, a negative number has a negative cube, and zero cubes to zero. Every number on the number line has just one cube root on the number line—not a positive-and-negative pair.
Fourth roots, fifth roots and beyond
We can reverse other powers too. Since 2 × 2 × 2 × 2 = 16, the fourth root of 16 is 2. With one more factor of 2, we get 32, so the fifth root of 32 is 2.
16 = 2Four equal factors make 16
32 = 2Five equal factors make 32
The small number is called the order of the root. An nth root reverses an nth power: multiply n copies of the answer to check it. A square root has order 2, usually left unwritten.
As with square roots, an even-order root symbol chooses the non-negative value.
On a calculator, use the cube-root key for cube roots, or the nth-root template for other orders. For 32, enter 5 in the small order slot and 32 under the root. The result is 2; checking 25 = 32 confirms that the entries are the right way round.
Why the order matters for negatives
Every extra negative factor changes the sign of a product. An even number of negative factors can all be paired off, giving a positive result. With an odd number, one negative factor is left over.
(−2) × (−2) × (−2) = −8
That's why an odd-order root can take a negative number: for example, −243 = −3. But no number on the number line has a negative even power, so −16 has no value on that line.
For a positive number, an even-order root symbol gives the non-negative root, just as a square-root symbol does. Both 24 and (−2)4 equal 16, but 16 means 2, not ±2.
Key points
- The order tells you how many equal factors to use. Check a root by raising it to that power.
- An odd-order root keeps the sign and gives one value on the number line.
- An even-order root symbol gives the non-negative root. A negative number has no even-order root on the number line.