simplify: 2 + sqrt -8, - sqrt -98, sqrt -216, - sqrt  -36, i sqrt -32, sqrt -128, (3i)(-4i)(7i), (-3i)-(3i), (2i)(-8i), (-4i)+(3i), -8i $$ \cdot $$ 4i $$ \cdot $$ i, (8i)^2, i^67, i^45, i^14, i^39300, i^8940, i^50, (-4 - 7i) - (-6+4i), (-5 - 4i) + (-3 + 2i), (4 + 5i)(-7 - 2i) + (1 - 5i) - (-1 + 7i), (-2i)(-5 + 3i)(3 - 8i), 2 sqrt -3 $$ \cdot $$ sqrt -12, (8+i)^2, (-4 + 5i)(-5 + 5i), (-6i)(-7 + 8i) + (i)(-i), -6 (-4i) + (2i)(8 + 5i), sqrt -10 $$ \cdot $$sqrt -12, (-8 + 7i) + (-4 + 8i) - (1 + 7i)

This is a collection of complex number expressions that involve both real and imaginary numbers. Let's go through each one and simplify them.

1. **$ 2 + \sqrt{-8} $**:  
   $ \sqrt{-8} = \sqrt{8} \cdot i = 2\sqrt{2}i $, so:
   \[
   2 + \sqrt{-8} = 2 + 2\sqrt{2}i
   \]

2. **$ -\sqrt{-98} $**:  
   $ \sqrt{-98} = \sqrt{98} \cdot i = 7\sqrt{2}i $, so:
   \[
   -\sqrt{-98} = -7\sqrt{2}i
   \]

3. **$ \sqrt{-216} $**:  
   $ \sqrt{-216} = \sqrt{216} \cdot i = 6\sqrt{6}i $, so:
   \[
   \sqrt{-216} = 6\sqrt{6}i
   \]

4. **$ -\sqrt{-36} $**:  
   $ \sqrt{-36} = 6i $, so:
   \[
   -\sqrt{-36} = -6i
   \]

5. **$ i \sqrt{-32} $**:  
   $ \sqrt{-32} = \sqrt{32} \cdot i = 4\sqrt{2}i $, so:
   \[
   i \sqrt{-32} = i \cdot 4\sqrt{2}i = -4\sqrt{2}
   \]

6. **$ \sqrt{-128} $**:  
   $ \sqrt{-128} = \sqrt{128} \cdot i = 8\sqrt{2}i $, so:
   \[
   \sqrt{-128} = 8\sqrt{2}i
   \]

7. **$ (3i)(-4i)(7i) $**:  
   Multiply the imaginary numbers:
   \[
   (3i)(-4i)(7i) = (3 \cdot -4 \cdot 7) \cdot i^3 = -84 \cdot (-i) = 84i
   \]

8. **$ (-3i) - (3i) $**:  
   \[
   (-3i) - (3i) = -6i
   \]

9. **$ (2i)(-8i) $**:  
   \[
   (2i)(-8i) = -16i^2 = 16
   \]

10. **$ (-4i) + (3i) $**:  
   \[
   (-4i) + (3i) = -i
   \]

11. **$ -8i \cdot 4i \cdot i $**:  
   \[
   (-8i)(4i)(i) = -32i^3 = 32i
   \]

12. **$ (8i)^2 $**:  
   \[
   (8i)^2 = 64i^2 = -64
   \]

13. **$ i^{67} $**:  
   Since $ i^4 = 1 $, we use $ i^{67} = i^{67 \mod 4} = i^3 = -i $.

14. **$ i^{45} $**:  
   $ i^{45} = i^{45 \mod 4} = i $.

15. **$ i^{14} $**:  
   $ i^{14} = i^{14 \mod 4} = i^2 = -1 $.

16. **$ i^{39300} $**:  
   $ i^{39300} = i^{39300 \mod 4} = i^0 = 1 $.

17. **$ i^{8940} $**:  
   $ i^{8940} = i^{8940 \mod 4} = i^0 = 1 $.

18. **$ i^{50} $**:  
   $ i^{50} = i^{50 \mod 4} = i^2 = -1 $.

19. **$ (-4 - 7i) - (-6 + 4i) $**:  
   Subtract complex numbers:
   \[
   (-4 - 7i) - (-6 + 4i) = (-4 + 6) + (-7i - 4i) = 2 - 11i
   \]

20. **$ (-5 - 4i) + (-3 + 2i) $**:  
   \[
   (-5 - 4i) + (-3 + 2i) = (-5 - 3) + (-4i + 2i) = -8 - 2i
   \]

21. **$ (4 + 5i)(-7 - 2i) + (1 - 5i) - (-1 + 7i) $**:  
   First, multiply $ (4 + 5i)(-7 - 2i) $:
   \[
   (4 + 5i)(-7 - 2i) = -28 -

Learn how to simplify complex number expressions involving square roots of negative numbers, powers of i, and operations between imaginary units. This guide provides detailed step-by-step solutions for students in Grades 10-12, covering key algebraic concepts related to complex numbers.

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