Simplify the given expression.
negative StartRoot negative 3 EndRoot StartRoot negative 5 EndRoot StartRoot negative 15 EndRoot
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Part 1
negative StartRoot negative 3 EndRoot StartRoot negative 5 EndRoot StartRoot negative 15 EndRootequals
  
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​(Simplify your answer. Type your answer in the form aplusbj​.)

Let's simplify the given expression:

\[
-\sqrt{-3} \times \sqrt{-5} \times \sqrt{-15}
\]

### Step 1: Simplify the square roots involving negative numbers.

We know that the square root of a negative number involves the imaginary unit \( i \), where:

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

Thus, we can express each square root with a negative number as:

\[
\sqrt{-3} = i\sqrt{3}, \quad \sqrt{-5} = i\sqrt{5}, \quad \sqrt{-15} = i\sqrt{15}
\]

### Step 2: Rewrite the expression with imaginary parts.

Substitute these into the expression:

\[
-\sqrt{-3} \times \sqrt{-5} \times \sqrt{-15} = -\left(i\sqrt{3}\right) \times \left(i\sqrt{5}\right) \times \left(i\sqrt{15}\right)
\]

### Step 3: Simplify the multiplication.

Multiply the constants and imaginary units separately:

\[
(-1) \times i \times i \times i = -i^3 = -(-i) = i
\]

Now multiply the square roots:

\[
\sqrt{3} \times \sqrt{5} \times \sqrt{15} = \sqrt{3 \times 5 \times 15} = \sqrt{225} = 15
\]

### Step 4: Combine everything.

The expression simplifies to:

\[
i \times 15 = 15i
\]

### Final Answer:
\[
15i
\]

---

Would you like to see more details on any part of the simplification, or have any questions? Here are a few related questions you might find helpful:

1. How do you simplify expressions involving square roots of negative numbers?
2. What is the property of \( i \) when multiplying it (e.g., \( i^2 \), \( i^3 \))?
3. How do you handle complex numbers in multiplication?
4. What happens when you multiply an imaginary number by a real number?
5. How do you simplify square roots involving variables? 

**Tip:** Always break down complex expressions involving square roots of negative numbers by factoring out \( i \) first, then simplify the real parts separately!

Simplify the given expression: negative StartRoot negative 3 EndRoot StartRoot negative 5 EndRoot StartRoot negative 15 EndRoot

Learn how to simplify expressions involving square roots of negative numbers with a detailed step-by-step solution. This problem uses imaginary numbers and square root properties to arrive at the final simplified expression 15i. Suitable for students in Grades 9-12.

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