Question

simplify: \\(\sqrt{-36} - \sqrt{-4} + \sqrt{-81}\\)

Explanation:

Step1: Recall imaginary unit

Recall that for a negative number \( -a \) (where \( a>0 \)), \( \sqrt{-a} = \sqrt{a}\cdot\sqrt{-1}=i\sqrt{a} \), where \( i \) is the imaginary unit with \( i^2=-1 \).
So, simplify each square - root term:

• For \( \sqrt{-36} \), we have \( \sqrt{-36}=\sqrt{36}\cdot\sqrt{-1} = 6i \) (since \( \sqrt{36} = 6 \) and \( \sqrt{-1}=i \)).
• For \( \sqrt{-4} \), we have \( \sqrt{-4}=\sqrt{4}\cdot\sqrt{-1}=2i \) (since \( \sqrt{4} = 2 \) and \( \sqrt{-1}=i \)).
• For \( \sqrt{-81} \), we have \( \sqrt{-81}=\sqrt{81}\cdot\sqrt{-1}=9i \) (since \( \sqrt{81}=9 \) and \( \sqrt{-1}=i \)).

Step2: Substitute and simplify

Substitute the simplified forms of the square - root terms into the original expression \( \sqrt{-36}-\sqrt{-4}+\sqrt{-81} \):
\( 6i-2i + 9i \)
Combine like terms. When combining like terms with the imaginary unit \( i \), we add or subtract the coefficients of \( i \).
The coefficient of \( i \) in \( 6i \) is 6, in \( - 2i \) is - 2, and in \( 9i \) is 9.
So, \( (6 - 2+9)i=(4 + 9)i = 13i \)

Answer:

\( 13i \)