Fully simplify.

left parenthesis, minus, 3, square root of, minus, 16, end square root, right parenthesis, left parenthesis, minus, 6, square root of, 64, end square root, right parenthesis

(−3−16​)(−664​)

Let's simplify the given expression step by step:

The expression is:  
\[
(-3 \sqrt{-16}) \times (-6 \sqrt{64})
\]

### Step 1: Simplify the square roots
- \(\sqrt{-16}\) involves a negative number under the square root. Using imaginary numbers, we know:
  \[
  \sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i
  \]
  where \(i\) is the imaginary unit.

- \(\sqrt{64}\) is straightforward:
  \[
  \sqrt{64} = 8
  \]

### Step 2: Substitute the simplified square roots into the expression
Now the expression becomes:
\[
(-3 \times 4i) \times (-6 \times 8)
\]

### Step 3: Perform the multiplications
- First, simplify the first part:
  \[
  -3 \times 4i = -12i
  \]
- Next, simplify the second part:
  \[
  -6 \times 8 = -48
  \]

### Step 4: Multiply the results
Now multiply the two results:
\[
(-12i) \times (-48) = 576i
\]

Thus, the fully simplified result is:
\[
\boxed{576i}
\]

Would you like more details or have any questions?

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Here are five related questions you might find useful:
1. How do you simplify expressions involving imaginary numbers?
2. What is the rule for multiplying two complex numbers?
3. How would you simplify \(\sqrt{-9}\)?
4. What is the difference between real and imaginary numbers?
5. How would you multiply complex numbers, like \((2 + 3i)(4 - i)\)?

**Tip:** When simplifying square roots of negative numbers, always remember to factor out the imaginary unit \(i\), since \(\sqrt{-1} = i\).

Learn to fully simplify the complex expression (−3√-16)(−6√64) with a step-by-step solution. Explore concepts like complex numbers, imaginary units, and properties of square roots, ideal for students in Grades 10-12.

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