At first, this problem seems rather easy. You merely need to divide these two values to get:

However, there are no answers that look like this! When this happens, you should consider rationalizing the denominator to eliminate the square root. This is a little more difficult than normal problems like this (ones that contain only the radical). However, if you complete a difference of squares in the denominator, you will be well on your way to having the right answer:

Carefully FOIL your denominator and distribute your numerator:

Look to your answers for an idea for factoring your numerator:

Note that 

This changes the initial equation to

For this equation to be valid, the exponents must be equal:

In order to solve for , the equation  must be written such that each set of exponents shares the same base:

Which like bases, it's now just a matter of solving for :

To find an equivalent, just multiply the top and bottom by the conjugate of the denominator.

Conjugate is the square root expression found in the denominator but with opposite sign.

So:

By simplifying, we get .

Since in all the answer choices have an integer in the denominator, we should multiply top and bottom by the conjugate of the denominator which is the square root expression with opposite sign.

So:

If we redistribute the negative, then the answer becomes 

.

To get into  form, multiply the fraction by bottom denominator's reciprocal.

The  is the numerator of the fraction and  is the denominator.

Final answer is . 

The greatest common divisor of 64 and 14 is 2, as found by the prime factorization of 64 and 14. The least common multiple of 16 and 52 is 208, which can be found by looking at the decimal when 52 is divided by 16. The remainder is 0.25, or ¹/₄ so the fourth multiple of 52 is 208, which is also divisible by 16.

Each of the numbers on the list must be able to "fit" (multiply evenly) into a larger number at the same time. I.e. the largest number (multiple) divided by any of the listed numbers will result in a whole number. For the coefficients, the maximum value is 10, and multiplying the highest two coefficients give us 60. Since 60 is divisible evenly by the lower values, we know that it is the least common multiple for the list. For the variables, both x and y will fit evenly into a theoretical number, "xy". We do not need an exponential version of this multiple as there are no exponents in the original list.

The least common multiple is the smallest number that is a multiple of all the numbers in the group.  Let's list some multiples of the two numbers and find the smallest number in common to both.

multiples of 45: 45, 90, 135, 180, 225, 270, ...

multiples of 60: 60, 120, 180, 240, 300, 360, ...

The smallest number in common is 180.

Factor each of our values into prime factors:

350 = 2 * 5² * 7

6270= 2 * 3 * 5 * 11 * 19

To find the least common multiple, we must choose the larger exponent for each of the prime factors involved.  Therefore, we will select 2, 5², and 7 from 350 and 3, 11, and 19 from 6270.

Therefore, our least common multiple is 2 * 3 * 5² * 7 * 11 * 19 = 219,450.