For small x, the first few terms of (1 + x)ⁿ give a good estimate, because the higher powers of x are tiny. (1 + x)¹⁰ ≈ 1 + 10x + 45x².
To estimate 1.01¹⁰, put x = 0.01: 1 + 0.1 + 0.0045 = 1.1045, which is very close to the true value 1.10462. For (2 − 3x)ⁿ, use the general term ⁿCᵣ 2ⁿ⁻ʳ(−3x)ʳ to find the first terms in ascending powers of x.
Worked example
Use the first three terms of (1 − x)⁷ with x = 0.01 to estimate 0.99⁷.
- (1 − x)⁷ ≈ 1 − 7x + 21x².
- 1 − 0.07 + 0.0021.
Answer:0.9321
Key words
- ascending powers
- x⁰, x¹, x², … in order
- approximation
- a close estimate
- small x
- makes higher terms negligible
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Common Core-referenced: HSA-APR.C.5