Question : The smallest among the numbers $2^{250},3^{150},5^{100}$, and $4^{200}$ is:
Option 1: $4^{200}$
Option 2: $5^{100}$
Option 3: $3^{150}$
Option 4: $2^{250}$
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Correct Answer: $5^{100}$
Solution :
The HCF of the exponentials 250, 150, 100, and 200 is 50.
So we can write:
$2^{250} = 2^{5\times 50} = (2^5)^{50} = 32^{50}$
$3^{150} = 3^{3\times 50} = (3^3)^{50}=27^{50}$
$5^{100} = 5^{2\times 50} = (5^2)^{50} = 25^{50}$
$4^{200} = 4^{4\times 50} = (4^4)^{50} = 256^{50}$
It can be seen that $25^{50}$ or $5^{100}$ is the smallest.
Hence, the correct answer is $5^{100}$.
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