# How do you find #sqrt(1521)# ?

##### 1 Answer

#### Explanation:

This link here lists several methods to compute square roots: https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

The digit-by-digit method is an extremely useful method to find exact answers but is extremely slow.

I will use an estimation method here. Don't worry if you do not understand everything in the following paragraph.

Newton's method states that the zeros of a function

#P# can be approximated as#x_1=x_0-(P(x_0))/(P'(x_0))# , where#x_0# is our initial guess,#x_1# is our next guess, and#P'# is the derivative of#P# with respect to#x# . Finding the square root of a number#A# is essentially finding a number#k# such that#k^2-A=0# . Substituting these in, we get#x_1=x_0-(x_0^2-A)/(2x_0)=(x_0^2+A)/(2x_0)# .

This means that, if we have a guess

Take note of three things. First, our new guesses will always be slightly larger than the actual answer (except when the guess is exactly the answer, then the new guess would be equal to the previous guess). We can round our final answer down to account for this. Second, some guesses may not work (for example, if our first guess is

Let's first try finding the square root of

Now, find the square root of