This complex numbers maze helps students practice operations with imaginary numbers in an engaging, puzzle-like format.

It looks like you've uploaded a worksheet titled "Complex Numbers: Complex Number Maze" that involves simplifying complex number expressions and navigating through a maze by choosing the correct simplified form.

Since I can't view images directly, I’ll guide you on how to solve this type of problem step-by-step based on typical content in such worksheets. If you can provide the specific expressions or describe the maze layout (e.g., starting point, options at each step), I can help solve it precisely.

However, here's how you would generally approach this kind of problem:

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🔷 Objective

Simplify each complex number expression using algebraic rules for complex numbers, particularly:
- $ i = \sqrt{-1} $
- $ i^2 = -1 $
- $ i^3 = -i $
- $ i^4 = 1 $
- $ \sqrt{-a} = i\sqrt{a} $ for $ a > 0 $

You will simplify expressions like:
- $ \sqrt{-9} = 3i $
- $ (2 + 3i)(1 - i) $ → Use FOIL
- $ \frac{1}{i} = -i $
- $ (a + bi)^2 $ → Expand

Then follow the path where each correct answer leads to the next step in the maze.

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🔷 Common Types of Problems in This Maze

Let’s go over common examples and their simplifications:

#### 1. Simplify $ \sqrt{-16} $
$$
\sqrt{-16} = \sqrt{-1 \cdot 16} = i\sqrt{16} = 4i
$$

#### 2. Simplify $ (3 + 2i)(1 - i) $
Use FOIL:
$$
= 3(1) + 3(-i) + 2i(1) + 2i(-i) = 3 - 3i + 2i - 2i^2 = 3 - i - 2(-1) = 3 - i + 2 = 5 - i
$$

#### 3. Simplify $ i^7 $
Note the cycle: $ i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 $, repeats every 4.
$$
i^7 = i^{4+3} = i^3 = -i
$$

#### 4. Simplify $ \frac{2}{i} $
Multiply numerator and denominator by the conjugate of denominator:
$$
\frac{2}{i} \cdot \frac{-i}{-i} = \frac{-2i}{-i^2} = \frac{-2i}{-(-1)} = \frac{-2i}{1} = -2i
$$

#### 5. Simplify $ (1 + i)^2 $
$$
= 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i
$$

#### 6. Simplify $ \sqrt{-25} $
$$
= 5i
$$

#### 7. Simplify $ 3i \cdot 4i $
$$
= 12i^2 = 12(-1) = -12
$$

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🔷 How to Solve the Maze

1. Start at the "Start" box.
2. Simplify the expression in the current cell.
3. Find the matching simplified answer among the adjacent boxes.
4. Move to that box and repeat until you reach "Finish."

For example:
- If one box has $ \sqrt{-4} $, simplify to $ 2i $.
- Then look for a neighboring box with $ 2i $ as its label or answer.
- Follow the chain.

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🔷 Tips

- Remember $ i^2 = -1 $ — use this constantly.
- For square roots of negative numbers: factor out $ i $.
- For multiplication: distribute carefully.
- For division: multiply by conjugate if needed.

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✔ Example Path (Hypothetical)

Suppose:
- Start: $ \sqrt{-9} $
- Simplify: $ 3i $
- Next box: $ 3i $ → leads to $ (3i)^2 $
- $ (3i)^2 = 9i^2 = -9 $
- Then find $ -9 $, etc.

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If you can copy and paste the actual expressions from the maze or describe the grid (e.g., what’s in each box), I can walk you through the exact solution step-by-step!

Alternatively, if you're looking for answers to specific problems on the sheet, list them and I’ll simplify them all.