Question

express in terms of i: (√(-12)) - (√(-3))
a -3i
b 3i
c 9i
d i√3
e i√9

Explanation:

Step1: Simplify \(\sqrt{-12}\)

Recall that \(\sqrt{-a}=\sqrt{a}\cdot i\) (where \(a>0\)). So, \(\sqrt{-12}=\sqrt{12}\cdot i\). Simplify \(\sqrt{12}\): \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\). Thus, \(\sqrt{-12}=2\sqrt{3}\,i\).

Step2: Simplify \(\sqrt{-3}\)

Using the same rule, \(\sqrt{-3}=\sqrt{3}\cdot i\).

Step3: Subtract the two terms

Now, \((\sqrt{-12})-(\sqrt{-3}) = 2\sqrt{3}\,i-\sqrt{3}\,i\). Combine like terms: \((2\sqrt{3}-\sqrt{3})i=\sqrt{3}\,i\)? Wait, no, wait. Wait, no, let's re - check. Wait, \(\sqrt{-12}=\sqrt{4\times(- 3)}=\sqrt{4}\times\sqrt{-3}=2\sqrt{-3}\). Oh! That's a simpler way. So \(\sqrt{-12}=2\sqrt{-3}\). Then \(\sqrt{-12}-\sqrt{-3}=2\sqrt{-3}-\sqrt{-3}=\sqrt{-3}\). But \(\sqrt{-3}=\sqrt{3}\,i\)? No, wait, no, earlier mistake. Wait, \(\sqrt{-12}=\sqrt{12}i = 2\sqrt{3}i\), \(\sqrt{-3}=\sqrt{3}i\). Then \(2\sqrt{3}i-\sqrt{3}i=\sqrt{3}i\)? But that's not one of the options. Wait, no, wait, maybe I made a mistake. Wait, the options have \(i\sqrt{3}\) as option D? Wait, the options: A: - 3i, B:3i, C:9i, D:\(i\sqrt{3}\), E:\(i\sqrt{9}\). Wait, \(\sqrt{9} = 3\), so E is \(3i\). Wait, no, let's re - do:

Wait, \(\sqrt{-12}=\sqrt{4\times(-3)}=\sqrt{4}\times\sqrt{-3}=2\sqrt{-3}\). Then \(\sqrt{-12}-\sqrt{-3}=2\sqrt{-3}-\sqrt{-3}=\sqrt{-3}\). But \(\sqrt{-3}=i\sqrt{3}\), which is option D? But wait, maybe my first approach was wrong. Wait, no, let's use the \(i\) definition correctly. \(\sqrt{-n}=i\sqrt{n}\) for \(n>0\). So \(\sqrt{-12}=i\sqrt{12}=i\sqrt{4\times3}=2i\sqrt{3}\), \(\sqrt{-3}=i\sqrt{3}\). Then \(2i\sqrt{3}-i\sqrt{3}=i\sqrt{3}\), which is option D? But wait, the option D is \(i\sqrt{3}\), option E is \(i\sqrt{9}=3i\). Wait, maybe I messed up the problem. Wait, no, wait the original problem: \((\sqrt{-12})-(\sqrt{-3})\). Let's compute numerically. Let \(i=\sqrt{-1}\). \(\sqrt{-12}=i\sqrt{12}\approx i\times3.464\), \(\sqrt{-3}=i\sqrt{3}\approx i\times1.732\). Then \(3.464i - 1.732i = 1.732i\), and \(\sqrt{3}\approx1.732\), so it's \(i\sqrt{3}\), which is option D. But wait, the option E is \(i\sqrt{9}=3i\), which is different. Wait, maybe I made a mistake in the problem reading. Wait, the problem is \((\sqrt{-12})-(\sqrt{-3})\). Let's factor out \(\sqrt{-3}\): \(\sqrt{-3}(\sqrt{4}-1)=\sqrt{-3}(2 - 1)=\sqrt{-3}=i\sqrt{3}\), so the answer is D. \(i\sqrt{3}\)

Answer:

D. \(i\sqrt{3}\)