Two to the hundred against ten to the thirty

Question

Which is larger, 2 to the power 100 or 10 to the power 30? Decide it in your head, without any written multiplication, and then estimate how many digits 2 to the 100 has and roughly what its leading digit is.

Accepted Answer

2 to the 100 is larger. It has 31 digits. The comparison. 2 to the 100 = (2 to the 10) to the 10 = 1,024 to the 10. And 10 to the 30 = (10 to the 3) to the 10 = 1,000 to the 10. Since 1,024 is bigger than 1,000, 2 to the 100 is bigger than 10 to the 30. How much bigger. The ratio is (1.024) to the 10. Since 1.024 to the 10 is about 1 + 10 × 0.024 plus a little, roughly 1.27, we get 2 to the 100 is about 1.27 × 10 to the 30. So it has 31 digits, and it starts with 1 (the exact value is 1,267,650,600,228,229,401,496,703,205,376). The rule of thumb. "2 to the 10 is about 10 to the 3" is one of t…

Two to the hundred against ten to the thirty

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