To solve the expression [tex]\(i^4 + \sqrt{-81} + i^2 + \sqrt{-36}\)[/tex] and rewrite it in the form [tex]\(a + bi\)[/tex], we will break down each component and evaluate it step by step.
First, we need to evaluate each term in the expression:
1. Evaluate [tex]\( i^4 \)[/tex]:
The imaginary unit [tex]\( i \)[/tex] is defined such that [tex]\( i = \sqrt{-1} \)[/tex].
[tex]\[
i^2 = -1
\][/tex]
[tex]\[
i^4 = (i^2)^2 = (-1)^2 = 1
\][/tex]
2. Evaluate [tex]\( \sqrt{-81} \)[/tex]:
Using the property of square roots of negative numbers, we can rewrite:
[tex]\[
\sqrt{-81} = \sqrt{81 \cdot (-1)} = \sqrt{81} \cdot \sqrt{-1} = 9i
\][/tex]
3. Evaluate [tex]\( i^2 \)[/tex]:
Using the definition of [tex]\( i \)[/tex]:
[tex]\[
i^2 = -1
\][/tex]
4. Evaluate [tex]\( \sqrt{-36} \)[/tex]:
Similarly:
[tex]\[
\sqrt{-36} = \sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1} = 6i
\][/tex]
Now, let's combine all these terms:
[tex]\[
i^4 + \sqrt{-81} + i^2 + \sqrt{-36} = 1 + 9i - 1 + 6i
\][/tex]
We can simplify this by combining like terms (real parts and imaginary parts separately):
- The real parts: [tex]\( 1 - 1 = 0 \)[/tex]
- The imaginary parts: [tex]\( 9i + 6i = 15i \)[/tex]
Thus, the expression simplifies to:
[tex]\[
0 + 15i
\][/tex]
Therefore, the correct answer in the form [tex]\(a + bi\)[/tex] is:
[tex]\[
0 + 15i
\][/tex]
The corresponding option is:
[tex]\[
\boxed{0+15i}
\][/tex]
