The problem asks us to simplify two expressions involving square roots of negative numbers using the imaginary unit $i$. The first expression is $\sqrt{-144}$, and the second is $\sqrt{-360}$.

1. Problem Description

The problem asks us to simplify two expressions involving square roots of negative numbers using the imaginary unit $i$. The first expression is $\sqrt{-144}$, and the second is $\sqrt{-360}$.

2. Solution Steps

For problem 39:

First, we rewrite the expression using the property $\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a}i$:

$\sqrt{-144} = \sqrt{144} \cdot \sqrt{-1}$

Since $\sqrt{144} = 12$ and $\sqrt{-1} = i$, we have:

$\sqrt{-144} = 12i$

For problem 40:

We rewrite the expression using the property $\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a}i$:

$\sqrt{-360} = \sqrt{360} \cdot \sqrt{-1}$

Now, we simplify $\sqrt{360}$. We can factor 360 as $360 = 36 \cdot 10$, so $\sqrt{360} = \sqrt{36 \cdot 10} = \sqrt{36} \cdot \sqrt{10} = 6\sqrt{10}$.

Therefore, $\sqrt{-360} = 6\sqrt{10}i$

3. Final Answer

For problem 39:

$12i$

For problem 40:

$6\sqrt{10}i$