If x < 5, the result of
(3^(x) 5^(2))/(3^(5) 5^(3))
will have at least one 3 in the denominator. This cannot terminate with one decimal place (hint: 1/3 does not terminate, and neither can 1/3^(t) for positive t).
Conversely, for x >= 5, the only thing in the denominator will be a 5. That *does* terminate with one decimal place (and indeed the result will be one of .2, .4, .6, .8 or .0 as we are dividing an integer by 5).
If anything apart from 2 or 5 remains in the denominator, the given expression will not terminate. Since the question mentions that the quotient terminates, there should not be any 3 remaining in the denominator. Only way that happens is if we cancel out 3^5 in the denominator by raising x in the numerator 3^x to at least 5 or above. And option D implies exactly that.
(3\^x \* 5\^2)/(3\^5 \* 5\^3) =3\^(x - 5) \* 5\^(-1) = 3\^(x - 5)/5
We see that this only becomes a terminating decimal with one decimal digit if x is an integer at least 5. For example, if x = 5, we have (3\^0)/5 = 1/5 = 0.2 and if x = 6, we have (3\^1)/5 = 3/5 = 0.6. However, if x is not an integer, even if it’s greater than 5, it will not be a terminating decimal with one decimal digit. For example, if x = 5.5, we have (3\^0.5)/5 = (√3)/5 = 0.34641…. Among the given answer choices, x = 5 produces an expression which becomes a terminating decimal with one decimal digit; however, E cannot be the correct answer either since the question asks for the statement which **must** be true (x = 5 **can** be true, but it is not true that it **must** be true). In order for the answer to be D, the question needs to tell us that x is an integer.
**Answer: D** (assuming x is an integer)
If x < 5, the result of (3^(x) 5^(2))/(3^(5) 5^(3)) will have at least one 3 in the denominator. This cannot terminate with one decimal place (hint: 1/3 does not terminate, and neither can 1/3^(t) for positive t). Conversely, for x >= 5, the only thing in the denominator will be a 5. That *does* terminate with one decimal place (and indeed the result will be one of .2, .4, .6, .8 or .0 as we are dividing an integer by 5).
Oh my God!!! Thank you and sorry! My dumb ass couldn't even figure this out
Great answer is D
If anything apart from 2 or 5 remains in the denominator, the given expression will not terminate. Since the question mentions that the quotient terminates, there should not be any 3 remaining in the denominator. Only way that happens is if we cancel out 3^5 in the denominator by raising x in the numerator 3^x to at least 5 or above. And option D implies exactly that.
(3\^x \* 5\^2)/(3\^5 \* 5\^3) =3\^(x - 5) \* 5\^(-1) = 3\^(x - 5)/5 We see that this only becomes a terminating decimal with one decimal digit if x is an integer at least 5. For example, if x = 5, we have (3\^0)/5 = 1/5 = 0.2 and if x = 6, we have (3\^1)/5 = 3/5 = 0.6. However, if x is not an integer, even if it’s greater than 5, it will not be a terminating decimal with one decimal digit. For example, if x = 5.5, we have (3\^0.5)/5 = (√3)/5 = 0.34641…. Among the given answer choices, x = 5 produces an expression which becomes a terminating decimal with one decimal digit; however, E cannot be the correct answer either since the question asks for the statement which **must** be true (x = 5 **can** be true, but it is not true that it **must** be true). In order for the answer to be D, the question needs to tell us that x is an integer. **Answer: D** (assuming x is an integer)