Binomial expansion for any power: approximations
Use the first terms of the binomial expansion to estimate roots and reciprocals of numbers near 1.
- أفغانستان: الصف ١١
- بريطانيا: السنة ١٣
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
For whole powers you know the binomial expansion: (١ + x)٣ = ١ + ٣x + ٣x٢ + x٣. The same formula works when the power n is a fraction or negative, but then it never ends.
(١ + x)n = ١ + nx + n(n − ١)٢!x٢ + …. When x is small, x² is much smaller than x, so the first few terms give a close answer.
Example: √١٫٠٤ = (١ + ٠٫٠٤)١٢ ≈ ١ + ٠٫٠٤٢ = ١٫٠٢. Just two terms, and the answer is very close.
Rule: when n is not a whole number the expansion is only valid for |x| < ١. Put the estimate on the zoomed line, type the coefficient, and draw where the expansion is valid. (This is A-level work in the UK.)