The expression is $\sqrt{9a^2}$, where $a < 0$.

Step 1: Simplify the expression

We can break this down as follows:

$\sqrt{9a^2} = \sqrt{9} \cdot \sqrt{a^2}$

Since $\sqrt{9} = 3$, and $\sqrt{a^2} = |a|$ (the absolute value of $a$), the expression becomes:

$3 \cdot |a|$

Step 2: Considering the condition $a < 0$

Since we are given that $a < 0$, the absolute value $|a|$ will be equal to $-a$ (because $a$ is negative, so $|a| = -a$).

Thus, the expression simplifies to:

$3 \cdot (-a) = -3a$

Final answer:

$\sqrt{9a^2} = -3a \quad \text{when} \quad a < 0$

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

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Here are 5 related questions that might interest you:

1. What happens to the expression $\sqrt{9a^2}$ if $a$ is positive?
2. How can we simplify $\sqrt{x^2}$ for any real number $x$?
3. What is the relationship between absolute value and square roots?
4. Can you explain the concept of square roots of negative numbers in terms of complex numbers?
5. How do we handle square roots of polynomial expressions, like $\sqrt{x^2 + 1}$?

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Tip: When simplifying expressions involving square roots, always pay attention to the signs of the variables and remember that $\sqrt{x^2} = |x|$.