# Is there an equation to find roots?

• Monkey Face
In summary, there are various mathematical equations and methods for finding square, cubic, and other roots of a number. These include methods such as using logarithms, Taylor series, and methods described in sources like Wikipedia. In the past, these methods were taught and used extensively, but with the advent of calculators and computers, they are less commonly used now. However, there has been recent progress in developing more efficient ways to find roots, particularly in areas such as cryptography and computer graphics.
Monkey Face
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

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.

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

## 1. How do you find the roots of an equation?

To find the roots of an equation, you can use the quadratic formula, factoring, or graphing techniques. The method you choose will depend on the type of equation you are working with.

## 2. What is the quadratic formula and how does it work?

The quadratic formula is a mathematical formula used to solve quadratic equations. It is written as x = (-b ± √(b²-4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. This formula works by finding the values of x that make the equation equal to zero.

## 3. Can any equation be solved using the quadratic formula?

No, the quadratic formula can only be used to solve quadratic equations, which are equations in the form of ax² + bx + c = 0. It cannot be used for equations with higher degrees or different forms.

## 4. Are there other methods besides the quadratic formula to find roots of an equation?

Yes, besides the quadratic formula, you can also use factoring and graphing techniques to find the roots of an equation. Factoring involves finding the common factors of an equation and setting them equal to zero. Graphing involves plotting the equation on a graph and finding the x-intercepts, which are the points where the equation crosses the x-axis.

## 5. Is there a single equation that can find roots of all types of equations?

No, there is not a single equation that can find roots of all types of equations. Different types of equations require different methods and techniques to find their roots. It is important to understand the characteristics and properties of the specific type of equation you are working with in order to find its roots accurately.

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