873595
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 $\displaystyle \sqrt{S}$,When choosing your initial value, 77 makes a good choice: $\displaystyle 77^2 = 5929 < 5955 < 78^2 = 6084$. Good luck!
- Choose an arbitrary positive start value $\displaystyle x_0$ (try to pick one close to the root).
- Let $\displaystyle x_{n+1}$ be the average of $\displaystyle x_n$ and $\displaystyle \frac S{x_n}$, i.e. $\displaystyle x_{n+1} = \frac12\left(x_n + \frac S{x_n}\right)$.
- Repeat steps 2 and 3 until you reach the desired accuracy.
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.