Question

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

Explanation:

Step1: Recall the imaginary unit definition

Recall that \( \sqrt{-a} = i\sqrt{a} \) for \( a > 0 \).
For \( \sqrt{-12} \), we have \( \sqrt{-12}=\sqrt{12\times(-1)} = i\sqrt{12} \). Simplify \( \sqrt{12} \): \( \sqrt{12}=\sqrt{4\times3}=2\sqrt{3} \), so \( \sqrt{-12} = 2i\sqrt{3} \).
For \( \sqrt{-3} \), we have \( \sqrt{-3}=i\sqrt{3} \).

Step2: Subtract the two expressions

Now, calculate \( (\sqrt{-12})-(\sqrt{-3}) \):
Substitute the values we found: \( 2i\sqrt{3}-i\sqrt{3} \).
Combine like terms: \( (2 - 1)i\sqrt{3}=i\sqrt{3} \). Wait, no, wait, let's re - check. Wait, \( 2i\sqrt{3}-i\sqrt{3}=(2 - 1)i\sqrt{3}=i\sqrt{3} \)? Wait, no, wait, \( \sqrt{12} = 2\sqrt{3} \), so \( \sqrt{-12}=i\sqrt{12}=2i\sqrt{3} \), and \( \sqrt{-3}=i\sqrt{3} \). Then \( 2i\sqrt{3}-i\sqrt{3}=i\sqrt{3} \)? Wait, but let's check the options. Option D is \( i\sqrt{3} \). Wait, but let's re - do the calculation.

Wait, \( \sqrt{-12}=\sqrt{4\times(-3)}=\sqrt{4}\times\sqrt{-3}=2\sqrt{-3} \). Then \( \sqrt{-12}-\sqrt{-3}=2\sqrt{-3}-\sqrt{-3}=\sqrt{-3}=i\sqrt{3} \). Yes, that's correct.

Answer:

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