Let's solve each part of the problem step by step.

(a) $\left( \sqrt{2} - i \right) - i\left( 1 - i\sqrt{2} \right) = -i2$

We will distribute and simplify.

1. Write the first expression: $\sqrt{2} - i$.
2. Expand $i(1 - i\sqrt{2})$: i(1 - i\sqrt{2}) = i - i^2\sqrt{2} = i + \sqrt{2} \quad \text{(since i^2 = -1)}.
3. Now subtract the two expressions: $\left( \sqrt{2} - i \right) - \left( i + \sqrt{2} \right).$ Simplifying: $\sqrt{2} - i - i - \sqrt{2} = -2i.$
4. So, we have $-2i$, which matches $-i2$.

Thus, part (a) is verified.

(b) $(2, -3)(-2, 1) = (-1, 8)$

This appears to be multiplication of two complex numbers represented in ordered pair form $(a, b)$ representing $a + bi$. We treat them as complex numbers and multiply them using the distributive property.

Let the two complex numbers be: $z_1 = 2 - 3i, \quad z_2 = -2 + i.$

Now multiply them: $(2 - 3i)(-2 + i) = 2(-2 + i) - 3i(-2 + i).$ Distribute both terms: $2(-2 + i) = -4 + 2i,$ $-3i(-2 + i) = 6i - 3i^2 = 6i + 3 \quad (\text{since } i^2 = -1).$ Now add the results: $-4 + 2i + 6i + 3 = (-4 + 3) + (2i + 6i) = -1 + 8i.$

Thus, part (b) is verified as $(-1, 8)$.

(c) $(3,1)(3,-1)\left( \frac{1}{5}, \frac{1}{10} \right) = (2,1)$

Again, we multiply these like complex numbers. Let’s work through it:

1. Start with $(3,1)(3,-1)$: $z_1 = 3 + i, \quad z_2 = 3 - i.$ Multiply them: $(3 + i)(3 - i) = 9 - i^2 = 9 + 1 = 10.$ So, $(3,1)(3,-1) = (10, 0)$ (because the imaginary part cancels out).

2. Now multiply by $\left( \frac{1}{5}, \frac{1}{10} \right)$: $(10, 0)\left( \frac{1}{5}, \frac{1}{10} \right).$ This means we multiply $10$ (the real part) by $\frac{1}{5}$, and the imaginary part $0$ by $\frac{1}{10}$: $10 \times \frac{1}{5} = 2, \quad 0 \times \frac{1}{10} = 0.$ Thus, the result is $(2, 1)$.

Therefore, part (c) is also verified.

All parts have been verified correctly.

Would you like any further explanations?

Here are 5 related questions to explore:

1. How do you multiply two complex numbers geometrically?
2. How can complex numbers be represented on an Argand diagram?
3. What is the significance of the imaginary unit $i$ in complex analysis?
4. How can division of complex numbers be performed?
5. How do polar coordinates relate to complex number multiplication?

Tip: When multiplying complex numbers, always remember to apply $i^2 = -1$ to simplify your result effectively.