This is a difficult but fast method which we mention especially because of its importance in the history of mathematics.
The method has to do with finding the values of x and y that satisfy the equation
x2 - Qy2 = 1.
This is the Brahmagupta-Bhaskara Equation, erroneously called Pell's Equation by Euler. The equation gives
This means that if we can find the values x and y, and if y happens to be large, then Q will be close to x2/y2 so that √Q will be approximated closely by x/y. The Chakravala method of Bhaskara uses a lemma of Brahmagrupta to achieve this.
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