If x is a positive integer, for how many different values of x is \sqrt{\frac{48}{x}} a whole number?

Explanation:

The fraction expression is:

You need to find the divisors of 48 that make 48/x a perfect square:

The perfect squares less than 48 are 1, 4, 9, 16, 25, and 36.

The divisors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Only 48/3 = 16, 48/12 = 4, and 48/48 = 1 are perfect squares.

Thus, 3, 12, and 48, are the values of x that make the expression a whole number.

Those are 3 numbers.

Hence, the answer is that there are 3 different values of x that make the fraction a whole number.

to find the total number of combinations, multiply the quantity of each item by each other:

2 pairs of shoes x 3 pairs of shorts = 6 different combinations.

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