Maths
2 to the 100 or 3 to the 63?
Which is bigger, 2¹⁰⁰ or 3⁶³? No calculator. Show me how you would decide.
- Hint 1Write both as powers with nearly the same exponent: 2¹⁰⁰ = 32²⁰ and 3⁶³ = 27²¹.
- Hint 2Compare (32/27)²⁰ with 27. The binomial expansion gives a lower bound.
- 2¹⁰⁰ = 32²⁰ and 3⁶³ = 27²¹ = 27 × 27²⁰.
- So 2¹⁰⁰ is bigger exactly when (32/27)²⁰ > 27, that is (1 + 5/27)²⁰ > 27.
- Every term of the binomial expansion is positive, so take the first six: 1 + 20(5/27) + 190(5/27)² + 1140(5/27)³ + 4845(5/27)⁴ + 15504(5/27)⁵.
- These are about 1 + 3.70 + 6.52 + 7.24 + 5.70 + 3.38 ≈ 27.5, already more than 27.
- So 2¹⁰⁰ > 3⁶³, though only just.
Where it lands: 2¹⁰⁰ is bigger.
The trap: Comparing 100 with 63 and the base 2 with 3 separately: the two effects pull opposite ways, so you need one combined comparison.
What a tutor might ask next: Using logarithms, roughly how many times bigger is it?
Now do one out loud. In the interview you think aloud with a tutor. Try a Maths problem with hints, follow-ups and Mia's feedback on how you think.