# Quick Estimation of Cube Roots

### Plus: Table of Cubes up to 200³

There are several ways to calculate cube roots by hand to any degree of accuracy that you desire. Common methods are the modified Babylonian algorithm and the long division technique. Unfortunately, these methods are arithmetic intensive. If you don't have a calculator with a cube root function and you need to estimate a radical, you can apply the rule described below. This formula yields an answer that is accurate to the tenths or hundredths place.

**Step 1:** First, call the number whose cube root you want to estimate M. Then, identify the perfect cubes on either side of M, and call the cube root of the smaller one C.

For example, suppose you want to estimate the cube root of 3000. Since 2744 = 14³ and 3375 = 15³, you have M = 3000 and C = 14. This tells you that the cube root of 3000 is somewhere between 14 and 15.

**Step 2:** To figure out where the cube root lies in between the two bounds, compute the ratio

[M - C³]/[(C+1)³ - C³] = [M - C³]/[3C² + 3C + 1]

and round to the nearest tenth or hundredth. Using M = 3000 and C = 14 in this example, you obtain

[3000 - 14³]/[3*14² + 3*14 + 1] = 256/631 ≈ 0.41 or 0.4.

**Step 3:** Add this number to C to obtain the estimate of the cube root. Continuing with the example above, you would estimate the cube root of 3000 as either 14.41 or 14.4.

As it turns out, the cube root of 3000 is about 14.422495703, so this estimation trick fairly accurate for simple applications.

If you need to cube numbers, use the table below.

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