The number i and Cartesian form Problem set A

Simplify each expression, giving the answer in the form \(x + iy\).

1. (a) \(i^{13}\)
2. (b) \(-6i^{3}\)
3. (c) \((4i)^2\)
4. (d) \((2i)^3\)

Answer each part.

1. (a) \(\sqrt{-81}\)
2. (b) \(\sqrt{-48}\)
3. (c) State the real part and the imaginary part of \(5 - \sqrt{-16}\).

Simplify each expression.

1. (a) \((5 - 2i) + (-3 + 9i)\)
2. (b) \((-1 + 6i) - (4 - 3i)\)

Expand and simplify, giving each answer in the form \(x + iy\).

1. (a) \((3 + 2i)(4 - 5i)\)
2. (b) \((-2 + i)(6 + 3i)\)

Expand and simplify.

1. (a) \((2 + 7i)^2\)
2. (b) \((4 - i)^2\)

Simplify each expression, giving the answer in the form \(x + iy\).

1. (a) \((3 - 2i)^2 - i(4 + 5i)\)
2. (b) \(2(1 + 3i)(2 - i) + 5i^3\)

Use your calculator to check complex arithmetic.

1. (a) Set the calculator to complex mode. Evaluate \((6 - i)(2 + 5i)\) and write the result in the form \(x + iy\).
2. (b) What does the calculator show for \(\sqrt{-1}\) in complex mode?

Karim writes \((3 + i)(3 - i) = 9 - i^2 = 8\). Dina writes the answer as \(10\).

1. (a) Which of them is right? Explain the mistake.
2. (b) \((6 + 5i) - (2 + 9i)\) Simplify, and say why no \(i^2\) step is needed.

In an alternating current circuit, two components are in series. Their impedances are \(Z_1 = 3 + 4i\) ohms and \(Z_2 = 2 - 7i\) ohms. The total impedance is \(Z = Z_1 + Z_2\). The current is \(I = 2 - i\) amps, and the voltage is \(V = IZ\).

1. (a) Find the total impedance \(Z\).
2. (b) Find the voltage \(V\) in the form \(x + iy\).

The powers of i repeat in a cycle of four.

1. (a) Find \(i + i^2 + i^3 + i^4\).
2. (b) Hence find \(i + i^2 + i^3 + \dots + i^{50}\).

This question uses a square to find a high power.

1. (a) Show that \((1 + i)^2 = 2i\).
2. (b) Hence find \((1 + i)^{8}\).