# Help with square root without calculator

• Jul 2nd 2008, 10:45 AM
MistaMista
Help with square root without calculator

I was not sure how to right this so i will also say it 87 radical 53595. I could do this with a calculator.
• Jul 2nd 2008, 11:35 AM
Reckoner
Quote:

Originally Posted by MistaMista

I was not sure how to right this so i will also say it 87 radical 53595. I could do this with a calculator.

First simplify a bit: $\displaystyle 87\sqrt{53595} = 87\sqrt{9\cdot5955} = 261\sqrt{5955}$

Now, there are many methods you could use to obtain a numerical approximation of $\displaystyle \sqrt{5955}$.

If you know calculus, you can use the Newton-Raphson method to find the zeros of the equation $\displaystyle x^2 - 5955 = 0$.

Alternatively, try the Babylonian method: For $\displaystyle \sqrt{S}$,
1. Choose an arbitrary positive start value $\displaystyle x_0$ (try to pick one close to the root).
2. Let $\displaystyle x_{n+1}$ be the average of $\displaystyle x_n$ and $\displaystyle \frac S{x_n}$, i.e. $\displaystyle x_{n+1} = \frac12\left(x_n + \frac S{x_n}\right)$.
3. Repeat steps 2 and 3 until you reach the desired accuracy.
When choosing your initial value, 77 makes a good choice: $\displaystyle 77^2 = 5929 < 5955 < 78^2 = 6084$. Good luck!
• Jul 2nd 2008, 11:50 AM
masters
Quote:

Originally Posted by MistaMista