Binomial theorem for rational and negative exponents (IB Math AA HL)
Updated October 2026
Quick answer: For any rational n, (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …, which is an infinite series that converges only for |x| < 1. To expand (a + bx)ⁿ, first take out aⁿ to get aⁿ(1 + bx/a)ⁿ; the expansion is then valid for |bx/a| < 1.
HL AA extends SL AA's binomial to (1+x)n for any real n, needing |x| < 1 for convergence. Paper 3 combines this with Maclaurin series.
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When n is a positive integer the series stops; for a negative or fractional n it goes on for ever.
Take out the constant first: (4 + x)1/2 = 2(1 + x/4)1/2, valid for |x| < 4.
Write out each coefficient with its brackets, such as (½)(−½)/2!, and simplify carefully: sign errors are the usual slip.
State the interval of validity, for example |x| < 4, as the question will usually ask for it.
To approximate a value, choose x small and inside the interval, so the first few terms are enough.
Worked example
Find the first three terms of the expansion of √(4 + x) in ascending powers of x, state the values of x for which it is valid, and use it to estimate √4.1.