Which expression is equivalent to [tex]\sqrt{-108} - \sqrt{-3}[/tex]?

A. [tex]5 i \sqrt{3}[/tex]
B. [tex]6 i \sqrt{3}[/tex]
C. [tex]7 i \sqrt{3}[/tex]
D. [tex]8 i \sqrt{3}[/tex]



Answer :

To determine which expression is equivalent to [tex]\( \sqrt{-108} - \sqrt{-3} \)[/tex]:

First, let's simplify each term separately using imaginary numbers.

The square root of a negative number can be expressed using the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].

1. Simplify [tex]\( \sqrt{-108} \)[/tex]:

[tex]\[ \sqrt{-108} = \sqrt{108} \cdot i \][/tex]

Next, factorize 108 under the square root:

[tex]\[ 108 = 36 \times 3 \][/tex]

So,

[tex]\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \][/tex]

Thus,

[tex]\[ \sqrt{-108} = 6\sqrt{3} \cdot i = 6i\sqrt{3} \][/tex]

2. Simplify [tex]\( \sqrt{-3} \)[/tex]:

[tex]\[ \sqrt{-3} = \sqrt{3} \cdot i = i\sqrt{3} \][/tex]

Now, we need to subtract these two results:

[tex]\[ \sqrt{-108} - \sqrt{-3} = 6i\sqrt{3} - i\sqrt{3} \][/tex]

Factor out the common term [tex]\( i\sqrt{3} \)[/tex]:

[tex]\[ \sqrt{-108} - \sqrt{-3} = (6-1)i\sqrt{3} = 5i\sqrt{3} \][/tex]

Therefore, the expression that is equivalent to [tex]\( \sqrt{-108} - \sqrt{-3} \)[/tex] is:

[tex]\[ \boxed{5i \sqrt{3}} \][/tex]