Greatest number among the 2⁵⁰, 3⁴⁰, 4³⁰ and 5²⁰ is
2019
Greatest number among the 2⁵⁰, 3⁴⁰, 4³⁰ and 5²⁰ is
- A.
5²⁰
- B.
2⁵⁰
- C.
3⁴⁰
- D.
4³⁰
Attempted by 2 students.
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Correct answer: C
To compare numbers written as different powers, rewrite each one so they share a common exponent — using the GCD of the exponents — since comparing values raised to the same power reduces to comparing their bases directly.
Find the GCD of the exponents 50, 40, 30 and 20, which is 10.
Rewrite 250 using exponent 10: 250 = (25)10 = 3210.
Rewrite 340 using exponent 10: 340 = (34)10 = 8110.
Rewrite 430 using exponent 10: 430 = (43)10 = 6410.
Rewrite 520 using exponent 10: 520 = (52)10 = 2510.
Compare the four bases obtained — 25, 32, 64 and 81 — the largest is 81, so 8110 is the largest value.
Cross-check using logarithms (base 10): 50 × log 2 ≈ 15.05; 40 × log 3 ≈ 19.08; 30 × log 4 ≈ 18.06; 20 × log 5 ≈ 13.98. The largest logarithm, ≈ 19.08, again corresponds to 340, confirming the result.
Therefore, 340 is the greatest of the four numbers.