Nth root

The nth root of a number $x$ is a value $y$ such that $y^n = x$, where $n$ is the degree and $x$ is the radicand. Finding the nth root is the inverse of exponentiation to the nth power in the sense that

$\sqrt[n]{x^n} = (x^n)^\frac{1}{n} = x$ .

A root can be written with the $\sqrt[]{}$ symbol or with a fractional exponent, for example $\sqrt[3]{125} = 125^\frac{1}{3}$.

When the radical symbol is written with no given radical, it is a square root (see below). Some examples of roots would be:

$\sqrt[2]{9}=3$

$\sqrt[2]{256} = 16$

$\sqrt[3]{64} = 4$

$\sqrt[10]{1024} = 2$

The second root is often called the square root, while the third root the cubic root. Negative roots may be defined if the radicand is not zero and the degree is odd. Even roots may be imaginary if the radicand is negative.

Solving equations with even positive integer powers results in two answers, one positive and one negative. For example, the equation

$x^{2}=9$

has two solutions

$\sqrt[2]{x^2} = x = \pm3$ since

$3^2=9$ and

$(-3)^2 = 9$ .

Babylonian Method - For Square Roots

This is the classical method for paper-pencil square rooting.

SQRT 7634169 = 

1. First group the number you are rooting into groups of 2 digits. If there are an odd number, have 1 digit left at the beginning

SQRT 7|63|41|69 =

2. Take the first group of digits(Farthest left. It would be the one digit if your number has an odd number of digits) and find a number that is the largest whole number possible that when squared, is less than or equal to the first group of number(s). That is the first digit in the solution. Squaredouble the answer so far and subtract from the group of number(s).

SQRT 7|63|41|69 = 2                                           7 - 4 =3

3. Bring the second pair of numbers out and place them at the end of the leftover number.

'SQRT 7|63|41|69 = 2                                           363   '                                                                                    

Suppose the answer so far is x and the number you just got is y. You have to find a number (n) so that (20x+n)n is less than or equal to y. Once again, try to make n as big as possible. n is now the second digit in the solution. Subtract (20x+n)n from y to get you new y and the solution so far is x. Repeat step 3 until you are satisfied.

SQRT 7|63|41|69 = 27        47 x 7 = 329             363 - 329 = 34

SQRT 7|63|41|69 = 27                                         3441

SQRT 7|63|41|69 = 276      546 x 6 = 3276         3441 - 3276 = 165

SQRT 7|63|41|69 = 276                                       16569

SQRT 7|63|41|69 = 2763    5523 x 3 = 16569     16569 - 16569 = 0