Binomial expansion for any power: approximations
Use the first terms of the binomial expansion to estimate roots and reciprocals of numbers near 1.
- UK: Year 13
- Afghanistan: grade 11
Guide
For whole powers you know the binomial expansion: (1 + x)3 = 1 + 3x + 3x2 + x3. The same formula works when the power n is a fraction or negative, but then it never ends.
(1 + x)n = 1 + nx + n(n − 1)2!x2 + …. When x is small, x² is much smaller than x, so the first few terms give a close answer.
Example: √1.04 = (1 + 0.04)12 ≈ 1 + 0.042 = 1.02. Just two terms, and the answer is very close.
Rule: when n is not a whole number the expansion is only valid for |x| < 1. 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.)