`sqrt(-5x^(8))- sqrt(-20x^(8)) + sqrt(-45x^(8))`, where x is a positive real number

Answer

To solve the expression \( \sqrt{-5x^8} - \sqrt{-20x^8} + \sqrt{-45x^8} \), where \( x \) is a positive real number, we will follow these steps:

### Step 1: Identify the square roots of negative numbers
We know that \( \sqrt{-a} = i\sqrt{a} \), where \( i = \sqrt{-1} \). Thus, we can rewrite each term in the expression:

\[ \sqrt{-5x^8} = i\sqrt{5x^8} \]
\[ \sqrt{-20x^8} = i\sqrt{20x^8} \]
\[ \sqrt{-45x^8} = i\sqrt{45x^8} \]

### Step 2: Simplify each square root
Next, we simplify each square root:

1. \( \sqrt{5x^8} = \sqrt{5} \cdot \sqrt{x^8} = \sqrt{5} \cdot x^4 \)
2. \( \sqrt{20x^8} = \sqrt{20} \cdot \sqrt{x^8} = \sqrt{20} \cdot x^4 = \sqrt{4 \cdot 5} \cdot x^4 = 2\sqrt{5} \cdot x^4 \)
3. \( \sqrt{45x^8} = \sqrt{45} \cdot \sqrt{x^8} = \sqrt{45} \cdot x^4 = \sqrt{9 \cdot 5} \cdot x^4 = 3\sqrt{5} \cdot x^4 \)

### Step 3: Substitute back into the expression
Now we can substitute these simplified terms back into the original expression:

\[ i(\sqrt{5} \cdot x^4) - i(2\sqrt{5} \cdot x^4) + i(3\sqrt{5} \cdot x^4) \]

### Step 4: Factor out common terms
We can factor out \( i \cdot x^4 \):

\[ i x^4 \left( \sqrt{5} - 2\sqrt{5} + 3\sqrt{5} \right) \]

### Step 5: Simplify the expression inside the parentheses
Now we simplify the expression inside the parentheses:

\[ \sqrt{5} - 2\sqrt{5} + 3\sqrt{5} = (1 - 2 + 3)\sqrt{5} = 2\sqrt{5} \]

### Step 6: Final expression
Putting it all together, we have:

\[ i x^4 \cdot 2\sqrt{5} = 2\sqrt{5} x^4 i \]

Thus, the final simplified form of the expression is:

\[ \boxed{2\sqrt{5} x^4 i} \]