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As we have the unwritten index $2$ for the sqare root, we multiply it by the index of the root inside the first root. $$\sqrt{\sqrt{x}} = \sqrt[2\times2]{x}= \sqrt[4]{x}$$ Another example is: $$\sqrt[3]{\sqrt[4]{x}} = \sqrt[3\times4]{x}= \sqrt[12]{x}$$
2. There are only two square roots of ii (as there are two square roots of any non-zero complex number), namely ± (1 + i) / √2. In the context of your answer, what happens is that the different values are e (πi / 2 + 2πik) / 2 = eπi / 4 + πik; but the value of this depends only on the parity of k, and so gives just two values, namely ± ...
So average these to get 1 2(17 12 + 24 17) = 577 408 = 1.4142156862⋯ This is already accurate to 6 decimal places. Repeating this process leads quickly to a good approximation of √2. One more iteration gives 1 2(577 408 + 2408 577) = 665857 470832 = 1.4142135623747⋯, which is accurate to about 12 digits.
Calculating the square root of a number is one of the first problems tackled with numerical methods, known I think to the ancient Babylonians. The observation is that if x, y> 0 and y ≠ √x then y, x / y will be on opposite sides of √x, and we could try averaging them. So try y0 = 1, yn + 1 = 1 2(yn + x yn).
By including the next term, we get 2–√ 00 = 14 + 17 − 11372 2 00 = 14 + 1 7 − 1 1372. This gives the superb result: 2–√ = 1.414212827... 2 = 1.414212827..., which is correct to 6 significant figures. APPENDIX - Another example of 13−−√ 13. Using the same approach and only the first two terms, we note that 36² = 1296.
This is of the form A = QΛQ − 1. If this is B2, then there will be a B of the form QΛ1 / 2Q − 1 (square this to check this is formally true). A square root of a diagonal matrix is just the square roots of the diagonal entries, so we have. B = (4 − 3 3 4)(√50 0 0 √25)(4 − 3 3 4) − 1. = 1 5(9 + 16√2 − 12 + 12√2 − 12 + 12 ...
One way is to convert the complex number into polar form. For z = reiθ, z2 = r2ei (2θ). So to take the square root, you'll find z1 / 2 = ± √reiθ / 2. Added: Just as with the nonnegative real numbers, there are two complex numbers whose square will be z. So there are two square roots (except when z = 0). Share.
What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "$987654321$" to $16$ decimal places in just $20$ iterations. I've now tried Newton's method as well as my own method (Newtons code as seen below) What is the fastest known algorithm for taking the second root of a number?
The square root of a negative number is possible! Square definition:"In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2".
Short answer: The Taylor series of √x at x0 = 0 does not exist because √x is not differentiable at 0. For any x0> 0, the Taylor series of √x at x0 can be computed using the Taylor series of √1 + u at u0 = 0. Long answer: The Taylor series of a function f that is infinitely differentiable at a point x0 is defined as.