Sure, let's solve the expression step by step.
We need to evaluate:
[tex]\[
-\sqrt{144} - \sqrt{-100} + \sqrt{16} + \sqrt{-1}
\][/tex]
### Step 1: Evaluate [tex]\(-\sqrt{144}\)[/tex]
[tex]\[
\sqrt{144} = 12 \quad \text{since} \quad 12^2 = 144
\][/tex]
Thus,
[tex]\[
-\sqrt{144} = -12
\][/tex]
### Step 2: Evaluate [tex]\(\sqrt{-100}\)[/tex]
The square root of a negative number involves imaginary numbers. By definition,
[tex]\[
\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1}
\][/tex]
We know that
[tex]\[
\sqrt{100} = 10 \quad \text{since} \quad 10^2 = 100
\][/tex]
And
[tex]\[
\sqrt{-1} = i \quad \text{(where \(i\) is the imaginary unit)}
\][/tex]
Thus,
[tex]\[
\sqrt{-100} = 10i
\][/tex]
### Step 3: Evaluate [tex]\(\sqrt{16}\)[/tex]
[tex]\[
\sqrt{16} = 4 \quad \text{since} \quad 4^2 = 16
\][/tex]
### Step 4: Evaluate [tex]\(\sqrt{-1}\)[/tex]
As previously mentioned,
[tex]\[
\sqrt{-1} = i
\][/tex]
### Step 5: Combine the terms
Now we combine the evaluated terms:
[tex]\[
-\sqrt{144} - \sqrt{-100} + \sqrt{16} + \sqrt{-1}
\][/tex]
Substituting the values we calculated:
[tex]\[
-12 - 10i + 4 + i
\][/tex]
### Step 6: Simplify the expression
Combine the real parts and the imaginary parts separately:
[tex]\[
(-12 + 4) + (-10i + i)
\][/tex]
Calculate the real part:
[tex]\[
-12 + 4 = -8
\][/tex]
Calculate the imaginary part:
[tex]\[
-10i + i = -9i
\][/tex]
### Final Expression:
Combining both parts, we get:
[tex]\[
-8 - 9i
\][/tex]
Therefore, the result of the expression is:
[tex]\[
\boxed{-8 - 9i}
\][/tex]
