Help with square root without calculator

**MistaMista** · July 2nd 2008, 10:45 AM

87

3595

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

**Reckoner** · July 2nd 2008, 11:35 AM

Originally Posted by MistaMista

87

3595

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: $87\sqrt{53595} = 87\sqrt{9\cdot5955} = 261\sqrt{5955}$

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

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

Alternatively, try the Babylonian method: For $\sqrt{S}$ ,

Choose an arbitrary positive start value $x_0$ (try to pick one close to the root).
Let $x_{n+1}$ be the average of $x_n$ and $\frac S{x_n}$ , i.e. $x_{n+1} = \frac12\left(x_n + \frac S{x_n}\right)$ .
Repeat steps 2 and 3 until you reach the desired accuracy.

When choosing your initial value, 77 makes a good choice: $77^2 = 5929 < 5955 < 78^2 = 6084$ . Good luck!

**masters** · July 2nd 2008, 11:50 AM

Originally Posted by MistaMista

87

3595

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

The best manual method I have found is discussed here: How to Find Square Roots Without a Calculator - by E. Oosterwal

It is much like long division. Follow the example given. Then try your radical. It's really quite simple. No guess work involved.

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