The number i and Cartesian form Problem set B

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

1. (a) \(i^{18}\)
2. (b) \(5i^{7}\)
3. (c) \((3i)^2\)
4. (d) \((5i)^3\)

Answer each part.

1. (a) \(\sqrt{-64}\)
2. (b) \(\sqrt{-75}\)
3. (c) State the real part and the imaginary part of \(-2 - \sqrt{-49}\).

Simplify each expression.

1. (a) \((-4 + 7i) + (8 - 3i)\)
2. (b) \((6 - 2i) - (-5 + 4i)\)

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

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

Expand and simplify.

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

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

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

Use your calculator to check complex arithmetic.

1. (a) Set the calculator to complex mode. Evaluate \((3 + 4i)(5 - 2i)\) and write the result in the form \(x + iy\).
2. (b) What does the calculator show for \(\sqrt{-25}\) in complex mode?

Ana writes \((5 + 2i)(5 - 2i) = 25 - 4i^2 = 21\). Ben writes the answer as \(29\).

1. (a) Which of them is right? Explain the mistake.
2. (b) \((-1 + 4i) - (2 + 3i)\) Simplify, and say why no \(i^2\) step is needed.

Two sensors report a signal as complex numbers, in millivolts: \(p = 4 - 3i\) and \(q = -1 + 8i\). The combined signal is \(s = p + q\). A filter multiplies the combined signal by the gain \(g = 3 + i\), so the output is \(gs\).

1. (a) Find the combined signal \(s\).
2. (b) Find the output \(gs\) 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 \(i + i^2 + i^3 + \dots + i^{47}\).

This question uses a square to find a high power.

1. (a) Show that \((1 - i)^2 = -2i\).
2. (b) Hence find \((1 - i)^{6}\).