https://file.aiquickdraw.com/mathbot/file/1729494700322olqcfch7.jpg

The expression shown in the image is:

\[
\sqrt{-4} \cdot \sqrt{-12}
\]

To solve this, we recognize that both square roots involve negative numbers, which introduces imaginary numbers. The square root of a negative number can be written as a multiple of \(i\), where \(i = \sqrt{-1}\).

### Step-by-step solution:

1. Express the square roots of negative numbers using \(i\):
   \[
   \sqrt{-4} = \sqrt{4} \cdot i = 2i
   \]
   \[
   \sqrt{-12} = \sqrt{12} \cdot i = \sqrt{4 \times 3} \cdot i = 2\sqrt{3} \cdot i
   \]

2. Now multiply the two terms:
   \[
   \sqrt{-4} \cdot \sqrt{-12} = (2i) \cdot (2\sqrt{3}i)
   \]

3. Multiply the constants and \(i\) terms:
   \[
   (2 \cdot 2\sqrt{3}) \cdot (i \cdot i) = 4\sqrt{3} \cdot (-1) = -4\sqrt{3}
   \]

So, the result of the expression is:
\[
-4\sqrt{3}
\]

Would you like more details or further clarifications on any step?

Here are some related questions for further practice:
1. How do you simplify expressions involving imaginary numbers?
2. What is the square root of a negative number in terms of \(i\)?
3. Can you explain why \(i^2 = -1\)?
4. How would you multiply two complex numbers?
5. What is the general form of a complex number?

**Tip:** Whenever you deal with square roots of negative numbers, always introduce the imaginary unit \(i\) before performing other operations.

Simplify the expression \( \sqrt{-4} \cdot \sqrt{-12} \)

Learn how to simplify the expression \( \sqrt{-4} \cdot \sqrt{-12} \) by applying the properties of imaginary numbers. This step-by-step solution covers key concepts in complex numbers, including the use of the imaginary unit \(i = \sqrt{-1}\). Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation