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The problem involves simplifying the complex expression \(\frac{8 + 3i}{1 + 7i}\) into the form \(a + bi\), where \(a\) and \(b\) are real numbers.

To simplify, we multiply both the numerator and denominator by the complex conjugate of the denominator, which is \(1 - 7i\). Here are the steps:

### Step 1: Multiply both numerator and denominator by \(1 - 7i\):
\[
\frac{8 + 3i}{1 + 7i} \times \frac{1 - 7i}{1 - 7i} = \frac{(8 + 3i)(1 - 7i)}{(1 + 7i)(1 - 7i)}
\]

### Step 2: Simplify the denominator using the difference of squares:
\[
(1 + 7i)(1 - 7i) = 1^2 - (7i)^2 = 1 - (-49) = 1 + 49 = 50
\]
So the denominator simplifies to 50.

### Step 3: Multiply the numerators:
\[
(8 + 3i)(1 - 7i) = 8(1) + 8(-7i) + 3i(1) + 3i(-7i)
\]
\[
= 8 - 56i + 3i - 21i^2
\]
Since \(i^2 = -1\), replace \(i^2\) with \(-1\):
\[
= 8 - 56i + 3i + 21 = (8 + 21) + (-56i + 3i) = 29 - 53i
\]

### Step 4: Combine the results:
\[
\frac{29 - 53i}{50} = \frac{29}{50} - \frac{53i}{50}
\]
Thus, the simplified expression is:
\[
\frac{29}{50} - \frac{53}{50}i
\]

### Final Answer:
\[
a + bi = \frac{29}{50} - \frac{53}{50}i
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you multiply complex numbers?
2. What is the conjugate of a complex number, and why is it useful?
3. How do you rationalize the denominator of a complex fraction?
4. What is the difference between the real and imaginary parts of a complex number?
5. How do you represent complex numbers on the complex plane?

**Tip:** Always remember that \(i^2 = -1\), a key property of imaginary numbers that simplifies many calculations.

Simplify (8 + 3i) / (1 + 7i) and give the answer in the form a + bi.

This page provides a detailed step-by-step solution to simplify the complex fraction (8 + 3i) / (1 + 7i), using the conjugate method and key concepts in imaginary numbers. Learn how to arrive at the final answer in the form a + bi, perfect for high school students in Grades 10-12.

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