Estimate 2^34 quickly. Key fact and value?

How to answer

Anchor on 2^10 ≈ 1000 (true 1024, ~2.4% high). 2^34 = (2^10)^3 · 2^4 ≈ 10^9 · 16 = 1.6×10^10. The 2.4%/decade error compounds over 3 decades to ~7.4% (1.024^3 ≈ 1.074): 1.6×10^10 × 1.074 ≈ 1.72×10^10. Exact is 1.718×10^10.

Key idea: Ignoring the 2.4%-per-decade upward drift of 1024≈1000; over many doublings it compounds and the raw estimate runs several percent low.