Approximate each square root to the nearest hundredth using the square algorithm
[tex] \sqrt{5} [/tex]
[tex] \sqrt{8} [/tex]
[tex] \sqrt{30} [/tex]
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Approximate each square root to the nearest hundredth using the square algorithm
[tex] \sqrt{5} [/tex]
[tex] \sqrt{8} [/tex]
[tex] \sqrt{30} [/tex]
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Approximating square roots using binomial expansion.
We want to (manually) approximate √2 by using the first few terms of the binomial series expansion of √1−2x=∞∑n=0(12n)(−2x)n|x|<12=1−x−12x2−12x3+⋯ Here we look for a way to determine appropriate values of x using the binomial expansion. In order to apply (1) we are looking for a number y with √1−2x=√2y2=y√2√2=1y√1−2x We see it is convenient to choose y to be a square number which can be easily factored out from the root. We obtain from (2) 1−2x=2y2x=12−y2 When looking for an appropriate y which fulfills (3) there are some aspects to consider: • We have to respect the radius of convergence |x|<12. • Since we want to calculate an approximation