Question

drag the tiles to the correct boxes to complete the pairs.
given that $x = 4 + 5i$ and $y = 2 - 9i$, match the equivalent expressions.
$53 - 26i$
$14 + 20i$
$2 - 32i$
$106 - 52i$
$x \cdot y$ $longrightarrow$
$4x - y$ $longrightarrow$
$-x + 3y$ $longrightarrow$
$2x \cdot y$ $longrightarrow$

Explanation:

Step1: Calculate $x \cdot y$

Multiply complex numbers:

$$\begin{align*} (4+5i)(2-9i)&=4\cdot2 + 4\cdot(-9i) + 5i\cdot2 + 5i\cdot(-9i)\\ &=8 - 36i + 10i - 45i^2\\ &=8 - 26i - 45(-1)\\ &=8 - 26i + 45\\ &=53 - 26i \end{align*}$$

Step2: Calculate $4x - y$

Substitute and simplify:

$$\begin{align*} 4(4+5i)-(2-9i)&=16 + 20i - 2 + 9i\\ &=(16-2)+(20i+9i)\\ &=14 + 29i \end{align*}$$

Step3: Calculate $-x + 3y$

Substitute and simplify:

$$\begin{align*} -(4+5i)+3(2-9i)&=-4 - 5i + 6 - 27i\\ &=(-4+6)+(-5i-27i)\\ &=2 - 32i \end{align*}$$

Step4: Calculate $2x \cdot y$

Use $x\cdot y=53-26i$:

$$ 2(53 - 26i)=106 - 52i $$

Answer:

$x \cdot y \longrightarrow 53 - 26i$
$4x - y \longrightarrow 14 + 29i$
$-x + 3y \longrightarrow 2 - 32i$
$2x \cdot y \longrightarrow 106 - 52i$