Simplify: `sqrt((-x)/(16))+sqrt((-x)/(25))-sqrt((-x)/(36))`, where 'x' is a positive real number.

Answer

To simplify the expression \( \sqrt{\frac{-x}{16}} + \sqrt{\frac{-x}{25}} - \sqrt{\frac{-x}{36}} \), where \( x \) is a positive real number, we will follow these steps:

### Step 1: Rewrite the square roots
Since \( x \) is a positive real number, \( -x \) is negative. We can express the square roots in terms of \( i \) (the imaginary unit), as follows:

\[ \sqrt{\frac{-x}{16}} = \sqrt{-1} \cdot \sqrt{\frac{x}{16}} = i \cdot \frac{\sqrt{x}}{4} \]

\[ \sqrt{\frac{-x}{25}} = \sqrt{-1} \cdot \sqrt{\frac{x}{25}} = i \cdot \frac{\sqrt{x}}{5} \]

\[ \sqrt{\frac{-x}{36}} = \sqrt{-1} \cdot \sqrt{\frac{x}{36}} = i \cdot \frac{\sqrt{x}}{6} \]

### Step 2: Substitute back into the expression
Now we can substitute these back into the original expression:

\[ \sqrt{\frac{-x}{16}} + \sqrt{\frac{-x}{25}} - \sqrt{\frac{-x}{36}} = i \cdot \frac{\sqrt{x}}{4} + i \cdot \frac{\sqrt{x}}{5} - i \cdot \frac{\sqrt{x}}{6} \]

### Step 3: Factor out \( i \sqrt{x} \)
We can factor out \( i \sqrt{x} \):

\[ = i \sqrt{x} \left( \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \right) \]

### Step 4: Find a common denominator
To simplify the expression inside the parentheses, we need a common denominator. The least common multiple of 4, 5, and 6 is 60. We can rewrite each fraction:

\[ \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60} \]

### Step 5: Combine the fractions
Now we can combine these fractions:

\[ \frac{15}{60} + \frac{12}{60} - \frac{10}{60} = \frac{15 + 12 - 10}{60} = \frac{17}{60} \]

### Step 6: Write the final expression
Now substituting back, we have:

\[ = i \sqrt{x} \cdot \frac{17}{60} = \frac{17 \sqrt{x}}{60} i \]

Thus, the simplified expression is:

\[ \frac{17 \sqrt{x}}{60} i \]