https://file.aiquickdraw.com/mathbot/file/17262148215708efgyvg2.jpg

The problem in the image asks you to simplify the square root of various perfect-square radicands. Let's break it down:

The expression shown is:

\[
\sqrt{4} \cdot \sqrt{b^2} \cdot \sqrt{c^2} \cdot \sqrt{3ab} = 2 \cdot \square \cdot c \cdot \sqrt{3ab}
\]

### Step-by-step simplification:

1. \(\sqrt{4} = 2\), since 4 is a perfect square.
2. \(\sqrt{b^2} = b\), since squaring and then taking the square root of \(b^2\) cancels out, leaving just \(b\).
3. \(\sqrt{c^2} = c\), by the same reasoning as above for \(b^2\).

After taking the square roots of the perfect squares, the equation simplifies to:

\[
2 \cdot b \cdot c \cdot \sqrt{3ab}
\]

So, the missing value in the box is \(b\).

The fully simplified expression is:

\[
2 \cdot b \cdot c \cdot \sqrt{3ab}
\]

Do you have any other questions or need further clarification?

### Related Questions:

1. How do you simplify square roots of non-perfect squares?
2. What is the general rule for simplifying square roots of products (e.g., \(\sqrt{a \cdot b}\))?
3. How would the equation change if one of the terms was not a perfect square?
4. Can you explain why \(\sqrt{b^2} = b\) and not \(-b\)?
5. What happens if the radicand includes negative terms, like \(\sqrt{-b^2}\)?

### Tip:
When simplifying square roots of products, you can split the square root across multiplication, i.e., \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), which helps in breaking down complex expressions.

Take the square root of the perfect-square radicand: sqrt(4) * sqrt(b^2) * sqrt(c^2) * sqrt(3ab) = 2 * [box] * c * sqrt(3ab).

Learn how to simplify expressions involving square roots of perfect-square radicands, including sqrt(4), sqrt(b^2), and sqrt(c^2). The problem illustrates the square root property and its application in algebraic expressions.

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