Question

Which property of multiplication is shown below? If x = 3 + 4i and y = 5 + 6i, x × y = y × x.

Answer 1

This is the commutative property.

Answer 2

Commutative property of multiplication

Explanation:

According to commutative property of multiplication , for two elements x and y , we have
`x* y=y* 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* 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* 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* y=15+18i+20i-24\\=\left ( 15-24 \right )+i\left ( 18+20 \right )\\=-9+38i`

`y* 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* y=y* x` .

Hence, commutative property is verified .