To determine for how many different values of [tex]\( x \)[/tex] will [tex]\( \sqrt{\frac{48}{x}} \)[/tex] be a whole number, let's go through the problem step by step:
1. Expression Simplification:
We need [tex]\( \sqrt{\frac{48}{x}} \)[/tex] to be a whole number. Let [tex]\( k \)[/tex] represent this whole number:
[tex]\[
\sqrt{\frac{48}{x}} = k
\][/tex]
Squaring both sides gives us:
[tex]\[
\frac{48}{x} = k^2
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{48}{k^2}
\][/tex]
For [tex]\( x \)[/tex] to be a positive integer, [tex]\( \frac{48}{k^2} \)[/tex] must also be an integer. Therefore, [tex]\( k^2 \)[/tex] must be a divisor of 48.
3. Find the Divisors:
Let's find the divisors of 48 that are perfect squares.
First, list the divisors of 48:
[tex]\[
1, 2, 3, 4, 6, 8, 12, 16, 24, 48
\][/tex]
Next, identify which of these are perfect squares:
[tex]\[
1, 4, 16
\][/tex]
4. Determine Values of [tex]\( x \)[/tex]:
- For [tex]\( k^2 = 1 \)[/tex]: [tex]\( x = \frac{48}{1} = 48 \)[/tex]
- For [tex]\( k^2 = 4 \)[/tex]: [tex]\( x = \frac{48}{4} = 12 \)[/tex]
- For [tex]\( k^2 = 16 \)[/tex]: [tex]\( x = \frac{48}{16} = 3 \)[/tex]
Each value of [tex]\( k^2 \)[/tex] gives us a valid [tex]\( x \)[/tex].
5. Conclusion:
The values of [tex]\( x \)[/tex] that make [tex]\( \sqrt{\frac{48}{x}} \)[/tex] a whole number are 48, 12, and 3. There are 3 such values.
Hence, the answer is [tex]\(\boxed{3}\)[/tex].
