Certainly! To simplify the given expression [tex]\((6+\sqrt{-16})-(2+\sqrt{-25})\)[/tex] and write it in the form [tex]\(a + bi\)[/tex], we'll follow these steps:
1. Simplify the square roots of the negative numbers:
Recall that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
[tex]\[
\sqrt{-16} = \sqrt{16 \cdot -1} = \sqrt{16} \cdot \sqrt{-1} = 4i
\][/tex]
[tex]\[
\sqrt{-25} = \sqrt{25 \cdot -1} = \sqrt{25} \cdot \sqrt{-1} = 5i
\][/tex]
2. Write the original complex numbers in the simplified form:
[tex]\[
6 + \sqrt{-16} = 6 + 4i
\][/tex]
[tex]\[
2 + \sqrt{-25} = 2 + 5i
\][/tex]
3. Perform the subtraction:
[tex]\[
(6 + 4i) - (2 + 5i)
\][/tex]
Subtract the real parts and the imaginary parts separately:
[tex]\[
(6 - 2) + (4i - 5i)
\][/tex]
4. Simplify the result:
[tex]\[
4 + (-1)i
\][/tex]
or simply:
[tex]\[
4 - i
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
4 - i
\][/tex]
This is in the form [tex]\(a + bi\)[/tex], where [tex]\(a = 4\)[/tex] and [tex]\(b = -1\)[/tex].
