Discussion
Which number amongst \(3^{31}\), \(4^{25}\), \(5^{21}\), \(9^{16}\) is the largest?
Answer: D
The largest is \(9^{16}\).
Compare the numbers in a common form. \(9^{16} = (3^2)^{16} = 3^{32}\), which is bigger than \(3^{31}\) by a factor of 3.
Take logarithms (base 10) for the other two:
\[ \begin{aligned} \log 3^{32} &= 32 \times 0.4771 \approx 15.27 \\ \log 4^{25} &= 25 \times 0.6021 \approx 15.05 \\ \log 5^{21} &= 21 \times 0.6990 \approx 14.68 \end{aligned} \]
So \(9^{16} \gt 4^{25} \gt 5^{21}\), and \(3^{31}\) is a third of \(9^{16}\). The trap is to assume the largest exponent wins; here the base matters as much as the power.
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