https://file.aiquickdraw.com/mathbot/file/1726097051836g3797sc3.jpg

Let's solve the given expression step by step:

The expression you need to multiply is:

\[
(\sqrt{-24}) (\sqrt{-7})
\]

### Step 1: Express in terms of imaginary numbers
We know that the square root of a negative number can be written in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\). Using this, we can simplify:

\[
\sqrt{-24} = \sqrt{24} \cdot i \quad \text{and} \quad \sqrt{-7} = \sqrt{7} \cdot i
\]

So, the expression becomes:

\[
(\sqrt{24} \cdot i) (\sqrt{7} \cdot i)
\]

### Step 2: Multiply the square roots and the imaginary parts
Now, multiply the terms:

\[
(\sqrt{24} \cdot \sqrt{7}) \cdot (i \cdot i) = \sqrt{24 \times 7} \cdot i^2
\]

Since \(i^2 = -1\), the expression becomes:

\[
\sqrt{168} \cdot (-1)
\]

### Step 3: Simplify the square root
We can simplify \(\sqrt{168}\) by factoring it:

\[
\sqrt{168} = \sqrt{4 \times 42} = \sqrt{4} \cdot \sqrt{42} = 2\sqrt{42}
\]

Thus, the expression simplifies to:

\[
-2\sqrt{42}
\]

### Final Answer:
The simplified form of the expression is:

\[
-2\sqrt{42}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How do you simplify square roots of larger numbers?
2. What are the steps to handle multiplication of imaginary numbers?
3. How do you factor square roots for simplification?
4. How does \(i^2 = -1\) impact complex number multiplication?
5. How do you handle expressions with multiple imaginary numbers?

**Tip:** Whenever you encounter the square root of a negative number, remember to rewrite it using \(i\) for a simpler calculation path.

Multiply: (√-24)(√-7). Simplify your answer and write it in the form a + bi.

This guide explains how to multiply complex numbers involving square roots of negative numbers, using properties of imaginary numbers. The step-by-step solution simplifies (√-24)(√-7) into -2√42. Suitable for high school students in Grades 10-12.

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