To solve the expression [tex]\( -\sqrt{49} - \sqrt{-2} - \sqrt{144} + \sqrt{-72} \)[/tex], let's evaluate each term step by step.
1. Evaluate [tex]\( -\sqrt{49} \)[/tex]:
[tex]\[
\sqrt{49} = 7, \text{ so } -\sqrt{49} = -7.
\][/tex]
2. Evaluate [tex]\( \sqrt{-2} \)[/tex]:
This involves the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].
[tex]\[
\sqrt{-2} = \sqrt{-1} \cdot \sqrt{2} = i\sqrt{2}.
\][/tex]
3. Evaluate [tex]\( -\sqrt{144} \)[/tex]:
[tex]\[
\sqrt{144} = 12, \text{ so } -\sqrt{144} = -12.
\][/tex]
4. Evaluate [tex]\( \sqrt{-72} \)[/tex]:
Similarly to the previous imaginary number calculation, the imaginary unit [tex]\( i \)[/tex] is involved.
[tex]\[
\sqrt{-72} = \sqrt{-1} \cdot \sqrt{72} = i \cdot \sqrt{72}.
\][/tex]
Further simplify [tex]\( \sqrt{72} \)[/tex]:
[tex]\[
\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.
\][/tex]
So,
[tex]\[
\sqrt{-72} = 6i\sqrt{2}.
\][/tex]
Now, combining all the evaluated parts together in the original expression [tex]\( -\sqrt{49} - \sqrt{-2} - \sqrt{144} + \sqrt{-72} \)[/tex], we have:
[tex]\[
-7 - i\sqrt{2} - 12 + 6i\sqrt{2}.
\][/tex]
Combine the real parts and the imaginary parts separately:
- Real parts: [tex]\( -7 - 12 = -19 \)[/tex].
- Imaginary parts: [tex]\( -i\sqrt{2} + 6i\sqrt{2} = 5i\sqrt{2} \)[/tex].
So the final expression in its simplest form is:
[tex]\[
-19 + 5i\sqrt{2}.
\][/tex]
This is the final result of the given mathematical expression.
