Answer: 64
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Explanation:
If n was negative, then the expression n/8 would be negative. However, the set of natural numbers is always positive. Specifically, the set of natural numbers is {1,2,3,4,5,...} which is also known as the counting numbers.
So we need the expression n/8 to be positive. That means n itself must be positive. The smallest possible value of n we could start with is n = 1. The problem though is that n/8 = 1/8 is not a natural number.
Something like n = 8 works though because n/8 = 8/8 = 1. Effectively, n must be a positive multiple of 8 for n/8 to be a natural number.
Let's try it with the third expression
[tex]\sqrt{n+225}=\sqrt{8+225} = \sqrt{233} \approx 15.264[/tex]
Since that's not a natural number, we must cross n = 8 off the list.
Through trial and error, you should find that the following other values of n do not work as well: 16, 24, 32, 40, 48, 56
Each of which are multiples of 8.
But something like n = 64 does work
[tex]\sqrt{n+225}=\sqrt{64+225} = \sqrt{289} = 17[/tex]
and this is the smallest value of n such that each of those three expressions yields a natural number. Therefore, n = 64 is the answer.
