simplify the expression to a + bi form: (7+7i)(-10+3i)

To simplify the expression \((7 + 7i)(-10 + 3i)\) into the form \(a + bi\), we need to use the distributive property (also known as the FOIL method for binomials). Here's the step-by-step process:

1. **Multiply each term in the first binomial by each term in the second binomial:**
   \[
   (7 + 7i)(-10 + 3i) = 7 \cdot (-10) + 7 \cdot 3i + 7i \cdot (-10) + 7i \cdot 3i
   \]

2. **Simplify each multiplication:**
   \[
   = -70 + 21i - 70i + 21i^2
   \]

3. **Combine like terms:**
   Since \(i^2 = -1\), substitute \(-1\) for \(i^2\):
   \[
   = -70 + 21i - 70i + 21(-1)
   \]
   \[
   = -70 + 21i - 70i - 21
   \]

4. **Further simplify by combining the real and imaginary parts:**
   \[
   = (-70 - 21) + (21i - 70i)
   \]
   \[
   = -91 - 49i
   \]

So, the simplified form of the expression \((7 + 7i)(-10 + 3i)\) is:

\[
-91 - 49i
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you multiply complex numbers using the FOIL method?
2. What is the significance of \(i^2 = -1\) in complex numbers?
3. How do you convert complex numbers from polar to rectangular form?
4. How can complex numbers be represented on the complex plane?
5. What are the applications of complex numbers in real-world scenarios?

**Tip:** Always remember to combine like terms carefully when dealing with complex numbers, keeping real and imaginary parts distinct!

Learn how to simplify the expression (7 + 7i)(-10 + 3i) into the form a + bi using the FOIL method. This problem involves complex numbers and the distributive property. Suitable for advanced high school students and learners interested in complex number operations.

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