How to find cube root of single digit numbers?

Question

I know methods of solving cube root of big numbers but they dont work with single digits. Suppose I need to find the cube root of 4 , How would I do that?

Answer 1

Suppose you have a super fast method for big number but for some unknown reason, it doesn't work on single digit.

Note that $$4000 = 4 \times 1000$$

$$4000^\frac13=4^\frac13\times 10$$

$$4^\frac13 = \frac{4000^\frac13}{10}$$

You can also consider binary search.

Answer 2

You can use logarithms if you want,

For eg :$Z = 4^{\frac13}$

$\ln Z= \frac13\ln4$

$\ln Z = \frac13(1.38629436112)$

$\ln Z = 0.46209812037$

$Z = e^{0.46209812037}$

$Z = 1.587401$

This works for any number ,big and small.Also I used a calculator to find $\ln$ values if you want to do it by hand , you can use a log book and use base 10 instead .

Answer 3

You can't really, at least not efficiently. If all you're allowed to do is division then your only option is trial and error: guess a cube root and divide your original number by that twice. If the result is too small then try a larger guess; if it's too large then try a smaller guess. Once you have upper and lower bounds for your guess you can then bisect the range each time (a.k.a. binary search) to narrow it down to the answer. But you'll be doing a lot of division.

For example let's consider the cube root of 5, and make an initial guess of 1.5. (1^3 = 1, 2^3 = 8, so seems reasonable?) Let's work to three decimal places. We can then divide 5 by 1.5 twice:

5 / 1.5 = 3.333, 3.333 / 1.5 = 2.222

If it were the cube root we'd get 1.5 again, but it's too big. So we need to divide by something larger:

5 / 1.6 = 3.125, 3.125 / 1.5 = 1.953 - result too big, try larger 5 / 1.7 = 2.941, 2.941 / 1.7 = 1.730 - result too big, larger still 5 / 1.8 = 2.778, 2.778 / 1.8 = 1.543 - result too small

We now have an upper bound, so the cube root is between 1.7 and 1.8. We then bisect this range:

5 / 1.75 = 2.857, 2.857 / 1.75 = 1.633 => try smaller: range is 1.7 to 1.75 => 1.725 5 / 1.725 = 2.899, 2.899 / 1.725 = 1.680 => smaller 5 / 1.713 = 2.919, 2.919 / 1.713 = 1.704 => smaller 5 / 1.707 = 2.929, 2.929 / 1.707 = 1.715 => larger 5 / 1.71 = 2.924, 2.924 / 1.71 = 1.710 => matches to 3 d.p.

So the cube root of 5 is 1.710 to three decimal places, using only long division. But that took us 18 divisions. (I'm glad I didn't have to do that by hand!)

If you really did need to do this by hand I'd suggest Newton-Raphson: it should converge a lot faster.