To perform the operation [tex]\((-2+\sqrt{-8})+(6-\sqrt{-54})\)[/tex], let's break it down step-by-step and then combine the parts.
1. Calculate [tex]\(\sqrt{-8}\)[/tex]:
- Since [tex]\(\sqrt{-8}\)[/tex] involves a negative number under the square root, we know we will get an imaginary number.
- The square root of [tex]\(-8\)[/tex] can be expressed as [tex]\(\sqrt{8} \cdot i\)[/tex].
- [tex]\(\sqrt{8}\)[/tex] simplifies to [tex]\(2\sqrt{2}\)[/tex].
- Therefore, [tex]\(\sqrt{-8} = 2\sqrt{2} \cdot i \approx 2.8284271247461903j\)[/tex].
2. Calculate [tex]\(\sqrt{-54}\)[/tex]:
- Similarly for [tex]\(\sqrt{-54}\)[/tex], it will also result in an imaginary number.
- The square root of [tex]\(-54\)[/tex] can be expressed as [tex]\(\sqrt{54} \cdot i\)[/tex].
- [tex]\(\sqrt{54}\)[/tex] simplifies to [tex]\(3\sqrt{6}\)[/tex].
- Therefore, [tex]\(\sqrt{-54} = 3\sqrt{6} \cdot i \approx 7.348469228349534j\)[/tex].
3. Form the two original complex numbers:
- The first complex number is [tex]\(-2 + \sqrt{-8}\)[/tex] which we now know is [tex]\(-2 + 2.8284271247461903j\)[/tex].
- The second complex number is [tex]\(6 - \sqrt{-54}\)[/tex] which we now know is [tex]\(6 - 7.348469228349534j\)[/tex].
4. Add the real parts:
- The real part of the first number is [tex]\(-2\)[/tex].
- The real part of the second number is [tex]\(6\)[/tex].
- Adding these gives: [tex]\(-2 + 6 = 4\)[/tex].
5. Add the imaginary parts:
- The imaginary part of the first number is [tex]\(2.8284271247461903j\)[/tex].
- The imaginary part of the second number is [tex]\(-7.348469228349534j\)[/tex].
- Adding these gives: [tex]\(2.8284271247461903j + (-7.348469228349534j) \approx -4.520042103603344j\)[/tex].
6. Combine the results:
- The resulting complex number is the sum of the real part and the imaginary part we calculated.
- Therefore, the result in standard form is: [tex]\(4 + (-4.520042103603344j)\)[/tex].
Ultimately, the final result is:
[tex]\[
4 + 10.176896353095724j
\][/tex]
