# Is there an equation to find roots?

Is there a mathematical equation to find square/cubic/etc. roots of a number?

Any help would be greatly appreciated - this is purely for my own interest. Also, I;m doing the C2 module of AS Maths so I may not understand more complex terms used in an explanation (if any are needed) - apologies in advance for my "ignorance". :)

Thanks!

~Monkey Face

## Answers and Replies

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
Could you explain very precisely what you mean by "find"?

I have the number 64. Other than by trial and error, or by simply knowning that the answer is 4, is there an equation that would give me that value of 4 as the cubic root?

I think you can get many equations that will give you the result of a cubic, square or whatever root you want. One of them is:

$$e^{\frac{1}{3}ln(64)}$$

If computers didn't exist, you could get the result from a table of logarithms.

Is there a way to do it by hand?

If you have a logarithm table or if you know one by memory, yes.
Other than that, the only other way I can remember is through a Taylor series expansion. See http://en.wikipedia.org/wiki/Taylor_series
With Taylor series you can only approach the value and it involves hard work and time.
I remember a teacher once saying hand calculators do it with Taylor series. Don't know if it is true.

Okay, thanks, I'll look into it.

Just as the square root can be found by hand, there is a method that is similar for finding the cube. http://www.itl.nist.gov/div897/sqg/dads/HTML/cubeRoot.html

Today, of course, nobody seems to bother with that stuff, but in my grandfather's day, I am 75, or probably to others in the days of your grandfather's father, they really used that method, or, anyway, they taught it down on the farm.

In those days the multiplication tables went to 12x12 because of the use of the dozen with eggs, etc., and the more difficult things were exercises--I assume, for the better students.

Much of that was taught in a one room schoolhouse--so my grandfather said. And, yes! They had to walk miles through the snow to get to the school in the Winter! Something considered inconceivable today.

mgb_phys
Science Advisor
Homework Helper
Ironically there has been a lot more progress in the last 20years in new ways to do quick arithmatic than the last few hundred.

Before calcualtors people needed to learn arithmetic tricks to make doing sums by hand easier (Feynman's biog lists the fun he had with them) - but the need (and \$) in making computer calculations faster mean that a lot of tricks for eg. finding roots more efficently have been developed recently. Main markets are cracking codes or for computer game graphics.

As for finding the cubic root of 64, you have to solve equation x^3=64