To solve \( \frac{9i}{8 + 8i} \), we multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \( 8 + 8i \) is \( 8 - 8i \).

\[
\frac{9i}{8 + 8i} \cdot \frac{8 - 8i}{8 - 8i} = \frac{9i(8 - 8i)}{(8 + 8i)(8 - 8i)}
\]

First, simplify the denominator:

\[
(8 + 8i)(8 - 8i) = 8^2 - (8i)^2 = 64 - (-64) = 64 + 64 = 128
\]

Now, simplify the numerator:

\[
9i(8 - 8i) = 9i \cdot 8 - 9i \cdot 8i = 72i - 72i^2 = 72i - 72(-1) = 72i + 72
\]

Thus, the expression becomes:

\[
\frac{72 + 72i}{128}
\]

Now, divide both terms by 128:

\[
\frac{72}{128} + \frac{72i}{128} = \frac{9}{16} + \frac{9i}{16}
\]

Therefore, the simplified form of \( \frac{9i}{8 + 8i} \) is:

\[
\frac{9}{16} + \frac{9i}{16}
\]

This is the final answer in the form of a complex number.

Do you need further details or have any questions?

Here are 5 related questions:
1. How do you find the complex conjugate of a number?
2. What is the modulus of a complex number?
3. Can you explain how multiplying by the conjugate helps in division of complex numbers?
4. How do you convert a complex number into polar form?
5. How do you perform addition and subtraction with complex numbers?

**Tip:** When dividing complex numbers, always multiply by the conjugate to simplify the expression.

Learn how to divide complex numbers using the example of 9i divided by (8+8i). This step-by-step solution explains the use of complex conjugates, multiplication, and simplification to reach the final answer. Suitable for students in Grades 10-12.

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