\sqrt{\left(3i\right)\cdot }\sqrt{\left(\frac{2}{2}i\right)}

To simplify the given expression \(\sqrt{(3i)} \cdot \sqrt{\left(\frac{2}{2}i\right)}\), let's follow these steps:

### Step 1: Simplify the terms inside the square roots
First, simplify \(\frac{2}{2}\):
\[
\frac{2}{2} = 1
\]
So the expression inside the second square root becomes \(1 \cdot i = i\).

Now the expression is:
\[
\sqrt{3i} \cdot \sqrt{i}
\]

### Step 2: Use the property of square roots
We can use the property of square roots which states \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

Applying this to the expression:
\[
\sqrt{3i} \cdot \sqrt{i} = \sqrt{(3i) \cdot (i)}
\]

### Step 3: Multiply the terms inside the square root
Now, multiply the terms inside the square root:
\[
(3i) \cdot (i) = 3i^2
\]

### Step 4: Simplify using the value of \(i^2\)
Recall that \(i^2 = -1\), so:
\[
3i^2 = 3(-1) = -3
\]

### Step 5: Take the square root of the result
Finally, take the square root of \(-3\):
\[
\sqrt{-3} = \sqrt{3} \cdot i
\]

### Final Answer
The simplified expression is:
\[
\sqrt{3} \cdot i
\]

Would you like any more details on this process or have any questions?

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Here are five related questions for further exploration:

1. How do you simplify expressions involving complex numbers?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the magnitude of a complex number?
4. Can you explain the concept of imaginary numbers in detail?
5. How does the square root of a negative number relate to complex numbers?

**Tip:** When dealing with complex numbers, remember that \(i^2 = -1\). This fundamental property is essential for simplifying expressions involving imaginary units.

Explore how to simplify the expression \(\sqrt{(3i)} \cdot \sqrt{\left(\frac{2}{2}i\right)}\) step-by-step. Learn about complex numbers, square roots, and the multiplication of complex numbers. Suitable for Grades 10-12.

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