Answer:
Commutative property of multiplication
Step-by-step explanation:
According to commutative property of multiplication , for two elements x and y , we have [tex]x\times y=y\times x[/tex] .
Here, [tex]x = 3+4 i , y= 5+6 i[/tex]
We will also use property : [tex](a+b)(c+d)=a(c+d)+b(c+d)[/tex]
[tex]x\times y=\left ( 3+4i \right )\left ( 5+6i \right )\\=3\left ( 5+6i \right )+4i\left ( 5+6i \right )[/tex]
Also, we know that [tex]i^2 = -1[/tex]
Therefore,
So,
[tex]x\times y=\left ( 3+4i \right )\left ( 5+6i \right )\\=3\left ( 5+6i \right )+4i\left ( 5+6i \right )\\=15+18i+20i+24i^2\\=15+18i+20i-24\\[/tex]
On adding real and imaginary parts separately , we get
[tex]x\times y=15+18i+20i-24\\=\left ( 15-24 \right )+i\left ( 18+20 \right )\\=-9+38i[/tex]
[tex]y\times x=\left ( 5+6i \right )\left ( 3+4i \right )\\=5\left ( 3+4i \right )+6i\left ( 3+4i \right )\\=15+20i+18i+24i^2\\=\left ( 15-24 \right )+i\left ( 20+18 \right )\\=-9+38i[/tex]
So, [tex]x\times y=y\times x[/tex] .
Hence, commutative property is verified .
