Cube Roots and Nth Roots: The Index Box Nobody Uses

Most people stop at the square root button and never touch the small index box beside it. That box is where the calculator earns its place, because the roots that turn up in real problems are rarely square ones.

Cube roots behave better than square roots

The cube root of 512 is 8. The cube root of 1,000 is 10. The cube root of 27 is 3, and the cube root of negative 27 is negative 3, which is the part that surprises people. Multiply three negatives together and the sign survives, so odd roots of negative numbers exist perfectly happily. Even roots do not, at least not among real numbers, which is why a square root calculator refuses a negative input.

Cube roots turn up whenever you have a volume and want a side length. A storage cube holding 27 cubic feet has three foot sides. A container holding 2,000 cubic centimetres, if it were a perfect cube, would measure about 12.6 centimetres on each edge, since 12.6 x 12.6 x 12.6 comes to 2,000.

A root is a fractional exponent wearing a different hat

The cube root of x is x to the power of one third. The fifth root is x to the power 0.2. If your calculator has an exponent key but no index box, 512 to the power (1/3) gets you the same 8. It also explains why roots and powers cancel: raise a number to the one third, then cube the result, and you are back where you started.

Nth roots are how growth rates get worked out

This is the use that matters outside a classroom. A pot of £10,000 becomes £18,000 after seven years. Divide to get the total growth factor, 1.8, then take the seventh root: 1.8 to the power of 1/7 is about 1.0876, so the annual rate is 8.76 percent. Not 80 divided by 7, which would be 11.4 percent and badly wrong, because each year's growth sits on top of the last. The mechanism is the one described in calculating compound interest, run backwards.

Doubling works the same way. Turning $5,000 into $10,000 in ten years needs the tenth root of 2, which is 1.0718. So 7.18 percent a year, every year, with nothing withdrawn.

Where the rule of 72 stops matching

The rule of 72 says a ten year double needs 72 divided by 10, or 7.2 percent. The true figure above was 7.18, so the shortcut holds. Stretch the horizon to 25 years and the rule says 2.88 percent while the twenty-fifth root of 2 gives 2.81. Close enough for a mental estimate, far enough out to matter on a large balance over a long period.

Using the root calculator

There are two inputs. The number you are taking the root of goes in the first field, and the index goes in the second: 2 for a square root, 3 for a cube root, 7 for a seven year growth rate. The index usually defaults to 2, so check it before you read the answer off the screen. Decimals are allowed in both boxes, so the fifth root of 0.5 returns 0.8706, which answers the question "what yearly factor halves something in five years".

Where the numbers come from depends on the job. For geometry it is a volume or an area you already measured. For finance it is an ending value divided by a starting value, with the index set to the number of years between the two. If you only ever need square roots, the dedicated square root calculator walkthrough goes further into perfect squares and simplifying surds.

Common questions

What is the cube root of 64? Four, because 4 x 4 x 4 = 64. The perfect cubes worth recognising on sight are 8, 27, 64, 125, 216, 343, 512 and 1,000.

Can you take the root of a negative number? With an odd index, yes. The cube root of negative 64 is negative 4. With an even index there is no real answer, only an imaginary one, so most calculators return an error rather than a number.

Is the fourth root the same as two square roots? Yes. The square root of 81 is 9, and the square root of 9 is 3, so 3 is the fourth root of 81. That trick works for any even index that factors into smaller steps.

Why does my answer show 7.9999999 instead of 8? Floating point rounding. A computer stores one third as a slightly truncated decimal, so cubing and un-cubing loses a sliver of precision along the way. If a result sits that close to a whole number, it is that whole number.