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