To solve the expression [tex]\(\sqrt{81} - \sqrt{-128} + \sqrt{100} + \sqrt{-8}\)[/tex], follow these steps:
1. Calculate the square roots of the positive numbers:
- [tex]\(\sqrt{81} = 9.0\)[/tex]
- [tex]\(\sqrt{100} = 10.0\)[/tex]
2. Calculate the square roots of the negative numbers:
- The square root of a negative number involves imaginary numbers where [tex]\(i\)[/tex] is the imaginary unit defined as [tex]\(i = \sqrt{-1}\)[/tex].
- [tex]\(\sqrt{-128} = \sqrt{128} \times i = 11.313708498984761i\)[/tex]
- [tex]\(\sqrt{-8} = \sqrt{8} \times i = 2.8284271247461903i\)[/tex]
3. Combine all parts of the expression:
- Now, our expression can be written as:
[tex]\[
9.0 - 11.313708498984761i + 10.0 + 2.8284271247461903i
\][/tex]
4. Separate the real parts and the imaginary parts:
- Real parts: [tex]\(9.0 + 10.0 = 19.0\)[/tex]
- Imaginary parts: [tex]\(-11.313708498984761i + 2.8284271247461903i = -8.485281374238571i\)[/tex]
5. Combine the real and imaginary parts:
- The combined result is:
[tex]\[
19.0 - 8.485281374238571i
\][/tex]
Therefore, the final result of the expression [tex]\(\sqrt{81} - \sqrt{-128} + \sqrt{100} + \sqrt{-8}\)[/tex] is:
[tex]\[
19 - 8.485281374238571i
\][/tex]
