To solve the expression [tex]\((9+\sqrt{-36})+(9+\sqrt{-49})\)[/tex] and simplify it into the form [tex]\(a + bi\)[/tex], follow these steps:
1. Identify the imaginary parts:
- Recall that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
- Therefore,
[tex]\[\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i\][/tex]
[tex]\[\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i\][/tex]
2. Rewrite the terms with imaginary numbers:
- Substitute the values back into the original expression:
[tex]\[(9 + 6i) + (9 + 7i)\][/tex]
3. Combine like terms:
- First, add the real parts:
[tex]\[9 + 9 = 18\][/tex]
- Next, add the imaginary parts:
[tex]\[6i + 7i = 13i\][/tex]
4. Form the result:
- Combine the results from the previous step:
[tex]\[18 + 13i\][/tex]
So, the simplified form of [tex]\((9+\sqrt{-36})+(9+\sqrt{-49})\)[/tex] is:
[tex]\[18 + 13i\][/tex]
