Practice questions

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7. Complex Numbers and Fractions

31. Simplify $\dfrac{4+3i}{5+3i}$. Write your answer in the form $a + bi$.

Solution:

Multiply the numerator and denominator by the complex conjugate of the denominator:

$=\dfrac{(4+3i)(5-3i)}{(5+3i)(5-3i)}$

Multiply out the numerator and remove the parentheses, and use the formula $(a+ib)(a-ib)=a^2+b^2$ for the denominator:

$=\dfrac{20-12i+15i-9i^2}{5^2+3^2}$

Substituting $i^2=-1$:

$=\dfrac{20+3i-9(-1)}{25+9}$

$=\dfrac{29+3i}{34}$

$=\dfrac{29}{34}+\dfrac{3i}{34}$

32. Multiply $2i(-4-i)^2$.

Solution:

$=2i(-(4+i))^2$

$=2i(4+i)^2$

Expanding using the formula $(a+b)^2 = a^2+2ab+b^2$:

$=2i(4^2+8i+i^2)$

Substitute $i^2=-1$:

$=2i(16+8i-1)$

$=2i(15+8i)$

$=30i+16i^2$

$=30i+16(-1)$

$=-16+30i$

33. Simplify.

(a) $i^{42}$

(b) $i^{17}$

(a).

Solution:

$i^{42}=i^2=-1$

Note that $2$ is the remainder when $42$ is divided by $4$.

(b).

Solution:

$i^{17}=i^1=i$

Note that $1$ is the remainder when $17$ is divided by $4$.