Roots and Radicals
If x is a positive integer, what is the least value of x for which
is an integer?
Read the question carefully. The variable x is an integer that makes the whole problem an integer. That seems a little confusing, but there are easy ways to think about this.
Identify the bottom line. x = ?
Assess your options. You could plug in your answers, but you will probably have to work the problem multiple times, which will waste time. You will be better off using logic to solve this problem. An integer is just a whole number, and in order to get a whole number from a radical, you will need a perfect square. You could list all the perfect squares that you know and try to get the numbers under the radical to equal them, but that will also take a certain amount of guesswork that will waste time. Instead, think systematically about how to get numbers out from under a radical sign.
Attack the problem. Instead of starting with the x, start with the first number under the radical. This number is a 3. The only way to get that 3 out from under the radical is to make it a perfect square. What is a perfect square involving 3? 3 * 3 = 9. You need another 3 in the top part of the fraction (the numerator) in order to get that 3 out from under the radical.
Then look at the 4 on the bottom part of the fraction. How can you cancel out this 4? The easiest way to that would be to make it equal 1! If you divide 4 by 4, the answer is 1. That means you also need a 4 in the numerator to cancel out that 4 in the denominator.
You have now determined that x must have the factor 3 and the factor 4. What is the smallest number with these two factors? 3 * 4 = 12. That means x = 12.
Loop back. You accounted logically for every number in your problem, so you are finished! Now is the time to select your answer and move on to the next problem. If you have extra time, you can double check the problem by working backwards from the answer. If you put x = 12 into the problem, you will find that there is a 9 under the radical, which means that the answer is 3, an integer. None of the other answer choices will work.
The correct answer is (D).
This is a hard question.