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Simplify the expression:

[tex]\[ -\sqrt{49} - \sqrt{-2} - \sqrt{144} + \sqrt{-72} \][/tex]


Sagot :

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.