Test 8 (Lessons 15-16): Complex Numbers Solutions

1. What is the definition of the imaginary number, i ?

The imaginary number i is defined as $i^2=–1$.

2. $i^{37}$

$$\begin{gathered} i^{36}·i \\ {\left(i^4\right)}^9·i \\ 1·i \end{gathered}$$

i

3. $i^{79}$

$$\begin{gathered} i^{76}·i^3 \\ {\left(i^4\right)}^{19}·i^3 \\ 1·\left(–i\right) \end{gathered}$$

– i

4. $\left(5i–36\right)+\left(25–8i\right)$

$–11–3i$

–11 + 3i
complex

5. $4\left(3+2i^2\right)$

$$\begin{gathered} 12+8i^2 \\ 12+8\left(–1\right) \end{gathered}$$

4
real complex

6. $\left(6i–3\right)\left(i^8+i\right)$

$$\begin{gathered} 6i^9+6i^2–3i^8–3i \\ \left(6·i^8·i\right)+6\left(–1\right)–\left(3·{\left(i^4\right)}^2\right)–3i \\ \left(6·{\left(i^4\right)}^2·i\right)–6–\left(3·1\right)–3i \\ \left(6·1·i\right)–6–3–3i \\ 6i–9–3i \end{gathered}$$

–9 + 3i
complex

1. $12i–\left(3i^3+5i^3\right)$

$$\begin{gathered} 12i–\left(8i^3\right) \\ 12i–8\left(–i\right) \end{gathered}$$

20i
pure imaginary complex

1. $\frac{3i}{\sqrt{–50}}$

$$\begin{gathered} \sqrt{–50}=i\sqrt{2·5^2}=5i\sqrt{2} \\ \frac{3i}{5i\sqrt{2}}\left(\frac{i\sqrt{2}}{i\sqrt{2}}\right) \\ \frac{3i^2\sqrt{2}}{5i^2\left(2\right)} \\ \frac{3\left(–1\right)\sqrt{2}}{5\left(–1\right)\left(2\right)} \end{gathered}$$

$\frac{3\sqrt{2}}{10}$

9. $\frac{–2i}{4–5i}$

$$\begin{gathered} \frac{–2i}{4–5i}\left(\frac{4+5i}{4+5i}\right) \\ \frac{–8i–10i^2}{16–25i^2} \\ \frac{–8i–10\left(–1\right)}{16–25\left(–1\right)} \end{gathered}$$

$\frac{10–8i}{41}$

10. Determine the value of M that forms a polynomial identity.
    ${\left(3x+i\right)}^2+\left(M–6ix\right)=\left(3x–2i\right)\left(3x+2i\right)$

M = 5