what is operation of finding a square root?

Answer 1

A square root of a is defined as a solution of the equation

x^2 - a = 0

over some field. Solving this equation, which operation we call "finding the square root", is an Algebraic problem.

Depending upon the field and the specific element a, therein, there might be no solution, one (that is, a unique) solution, a pair (that is, conjugate) solutions, or more. If a is an integer and a solution exists, then it is either an integer or an algebraic irrational number. In the field of real numbers, there are a pair of conjugate solutions if a > 0, one solution if a = 0, and none if a < 0. In the field of complex numbers, to which we will confine ourselves, henceforth, there always exist a pair of roots, whose sum is zero. They are complex-conjugate iff (= if and only if) the original number is real-complex and negative.

Finding a square-root is just a special case of solving a polynomial equation. Hence, we provide some background information on polynomials

Answer 2

By definition, square root is an operator that gives you the number which when squared (multiplied by itself), gives you back to original number. So sqrt(9) = 3, and sqrt(16) = 4, etc.

If you're asking how do we go about calculating the square root of something stranger like "10", which has to be a number somewhere in between 3 and 4 and thus a decimal, basically it's trial and error:

1) start with a number for your guess of the square root
2) square it
3) Compare your answer from #2 with the original number you're trying to find the square root of.
4) If your answer is too high, try again with a lower number. If the answer is too low, try again with a higher number.

For example, if we were finding the square root of 10, 3.5 squared gives us 12.25. That's higher than 10. But if we try 3.1 squared, we get 9.61. Trying 3.2 we get 10.24, so we know the answer is between 3.1 and 3.2. So we might try 3.15 next and so on. We'll never get to an exact value (you can mathematically prove that the square root of 10 is an irrational number, and this the decimals won't have an end). But calculating more and more gives you a closer and closer answer. Similarly, calculators use complicated math formulas involving addtion and subtraction that give you an approximate answer for square roots.

Answer 3

The most common method of square root calculation by hand is known as the "Babylonian method". It involves a simple algorithm, which will bring you closer and closer to the actual square root each time it is repeated. To find r, the square root of a real number x:

Start with an arbitrary positive start value r (the closer to the square root of x, the better).
Replace r by the average between r and x / r. (It is sufficient to take an approximate value of the average, not too close to the previous value of r and x / r in order to ensure convergence.)
Repeat steps 2 and 3.

Answer 4

The square root of x is x to the 1/2. Cube root is to the 1/3, and so on.

Answer 5

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