`sqrt((-x)/(4)) + sqrt((-x)/(16))- sqrt((-x)/(64))`, where x is a positive real number

Answer

To solve the expression \( \sqrt{\frac{-x}{4}} + \sqrt{\frac{-x}{16}} - \sqrt{\frac{-x}{64}} \), where \( x \) is a positive real number, we can follow these steps:

### Step 1: Rewrite the square roots
We start by rewriting each term in the expression using the property of square roots:

\[ \sqrt{\frac{-x}{4}} = \frac{\sqrt{-x}}{\sqrt{4}} = \frac{\sqrt{-x}}{2} \]

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

\[ \sqrt{\frac{-x}{64}} = \frac{\sqrt{-x}}{\sqrt{64}} = \frac{\sqrt{-x}}{8} \]

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

\[ \frac{\sqrt{-x}}{2} + \frac{\sqrt{-x}}{4} - \frac{\sqrt{-x}}{8} \]

### Step 3: Find a common denominator
The common denominator for \( 2, 4, \) and \( 8 \) is \( 8 \). We can rewrite each term with this common denominator:

\[ \frac{4\sqrt{-x}}{8} + \frac{2\sqrt{-x}}{8} - \frac{\sqrt{-x}}{8} \]

### Step 4: Combine the terms
Now we can combine the fractions:

\[ \frac{4\sqrt{-x} + 2\sqrt{-x} - \sqrt{-x}}{8} = \frac{(4 + 2 - 1)\sqrt{-x}}{8} = \frac{5\sqrt{-x}}{8} \]

### Step 5: Express \(\sqrt{-x}\) in terms of \(i\)
Since \( x \) is a positive real number, \( -x \) is negative. We can express \( \sqrt{-x} \) as:

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

### Step 6: Substitute back into the expression
Substituting this into our expression gives:

\[ \frac{5\sqrt{-x}}{8} = \frac{5\sqrt{x} \cdot i}{8} \]

### Final Answer
Thus, the final simplified form of the expression is:

\[ \frac{5\sqrt{x} \cdot i}{8} \]

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