Cube Root of a Number

What does it mean to find the cube root of a number?

Well, when we ask for the cube root of a number, we are really asking, "what number when multiplied by itself twice gives the number we want the cube root of".

For example, what is the cube root of the number 729? The real question we must answer is, "what number, when multiplied by itself twice, gives us the value 729"? While we probably can’t answer that question quickly of the top of our heads, it doesn’t take a lot of thought to come up with the answer: the answer is 9, because 9 x 9 x 9 = 729. So, the cube root of 729 is 9.

That’s not really too difficult. But suppose we want to know the cube root of 12; that is, we want to know what number, when multiplied by itself twice, equals 12? That’s not quite so easy. Most of us would use a calculator to find that answer. Unfortunately, most calculators do not have a key with a cube root symbol, , as we have with the square root key with the symbol on it, on the calculator. There are, of course, other methods that allow us to use a calculator to find a cube root, but we will not explore those methods in this article.

We could find cube root values using a trial-and-error procedure but that takes quite a bit of computational work. Suppose we want the cube root of 12, for example. We might start by trying 3.0 for the cube root. 3.0 x 3.0 x 3.0 = 27.0, so 3.0 is too large. We next might try 2.5 which gives us 2.5 x 2.5 x 2.5 = 15.626. So, 2.5 is also too large, so we might next try 2.2 as our trial value. 2.2 x 2.2 x 2.2 = 10.648, which is closer to 12 but too small. Let’s next try 2.29. 2.29 x 2.29 x 2.29 = 12.008989 which is almost the exact value for the cube root of 12 that we are looking for. If we continue our trial-and-error procedure, we can get our value for the cube root of 12 as close to its correct value as we wish. Using a calculator would tell us that , correct to as many decimal places as we wish by using as many trials as necessary to achieve the accuracy we want.

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