Calculate (60i-66i^2) ÷ 102i

i is iota 

We are given the expression:

\[
\frac{60i - 66i^2}{102i}
\]

### Step 1: Simplify the numerator
Recall that \(i^2 = -1\). Using this, we can simplify the expression:

\[
60i - 66i^2 = 60i - 66(-1) = 60i + 66
\]

Now the expression becomes:

\[
\frac{60i + 66}{102i}
\]

### Step 2: Separate into two fractions
We can split this into two simpler fractions:

\[
\frac{60i}{102i} + \frac{66}{102i}
\]

### Step 3: Simplify each fraction

1. The first fraction \(\frac{60i}{102i}\):
   \[
   \frac{60i}{102i} = \frac{60}{102} = \frac{5}{8}
   \]

2. The second fraction \(\frac{66}{102i}\):
   \[
   \frac{66}{102i} = \frac{66}{102} \cdot \frac{1}{i} = \frac{11}{17} \cdot \frac{-i}{i^2} = -\frac{11i}{17}
   \]
   Since \(\frac{1}{i} = -i\) (as \(\frac{1}{i} = \frac{-i}{i^2} = -i\)).

### Step 4: Final expression

Thus, the simplified form of the expression is:

\[
\frac{5}{8} - \frac{11i}{17}
\]

This is the final result.

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Would you like more details on any step, or further clarification?

Here are 5 related questions to explore:
1. How do you simplify complex expressions involving powers of \(i\)?
2. What is the significance of separating complex fractions?
3. Can you explain the process of rationalizing a denominator with \(i\)?
4. How do complex numbers behave in division?
5. Why is \(i^2 = -1\) fundamental to complex number calculations?

**Tip:** When dealing with complex numbers, always look for opportunities to simplify using \(i^2 = -1\), which can turn terms with \(i\) into real numbers.

Explore how to simplify the complex expression (60i - 66i^2) ÷ 102i with step-by-step guidance. This solution involves concepts of complex numbers, rationalizing denominators, and properties of the imaginary unit i. Suitable for Grades 10-12 students.

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