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 $\sqrt{-360}$.

AlgebraComplex NumbersImaginary UnitSquare RootsSimplification
2025/3/7

1. Problem Description

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

2. Solution Steps

For problem 39:
First, we rewrite the expression using the property a=a1=ai\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a}i:
144=1441\sqrt{-144} = \sqrt{144} \cdot \sqrt{-1}
Since 144=12\sqrt{144} = 12 and 1=i\sqrt{-1} = i, we have:
144=12i\sqrt{-144} = 12i
For problem 40:
We rewrite the expression using the property a=a1=ai\sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a}i:
360=3601\sqrt{-360} = \sqrt{360} \cdot \sqrt{-1}
Now, we simplify 360\sqrt{360}. We can factor 360 as 360=3610360 = 36 \cdot 10, so 360=3610=3610=610\sqrt{360} = \sqrt{36 \cdot 10} = \sqrt{36} \cdot \sqrt{10} = 6\sqrt{10}.
Therefore, 360=610i\sqrt{-360} = 6\sqrt{10}i

3. Final Answer

For problem 39:
12i12i
For problem 40:
610i6\sqrt{10}i