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Let's simplify the given expression into the form \( a + bi \). The expression is:

\[
\sqrt{49} + \sqrt{-27} + \sqrt{36} + \sqrt{-27}
\]

### Step-by-Step Simplification:

1. **Simplify real square roots:**
   \[
   \sqrt{49} = 7
   \]
   \[
   \sqrt{36} = 6
   \]

2. **Simplify the square roots of negative numbers:**
   The square root of a negative number involves imaginary numbers. Specifically, \( \sqrt{-n} = i\sqrt{n} \), where \( i \) is the imaginary unit (\( i = \sqrt{-1} \)).

   \[
   \sqrt{-27} = i\sqrt{27} = i \cdot 3\sqrt{3} = 3i\sqrt{3}
   \]
   There are two instances of \( \sqrt{-27} \), so both simplify to \( 3i\sqrt{3} \).

3. **Combine like terms:**
   The expression now becomes:
   \[
   7 + 6 + 3i\sqrt{3} + 3i\sqrt{3}
   \]

   Combine the real and imaginary parts:
   \[
   (7 + 6) + (3i\sqrt{3} + 3i\sqrt{3}) = 13 + 6i\sqrt{3}
   \]

Thus, the expression simplified to the form \( a + bi \) is:

\[
\boxed{13 + 6i\sqrt{3}}
\]

Would you like more details or have any questions?

Here are some related questions to explore:
1. What are the properties of imaginary numbers?
2. How do you add and subtract complex numbers?
3. How can you simplify square roots of negative numbers?
4. What are some practical uses of complex numbers in physics or engineering?
5. How do complex numbers behave in polar form?

**Tip:** When dealing with square roots of negative numbers, always factor out the \( i \) first to simplify the expression into the real and imaginary components.

Simplify the expression to a + bi form: √49 + √(-27) + √36 + √(-27)

Learn how to simplify the expression √49 + √(-27) + √36 + √(-27) into the form a + bi. This problem covers key concepts in complex numbers, including the square root of negative numbers and imaginary units. Suitable for students in Grades 10-12.

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