Question

Which equation illustrates the identity property of multiplication?

• A. $(\mathrm{a}+\mathrm{bi})\times \mathrm{c}=(\mathrm{ac}+\mathrm{bci})$
• B. $(\mathrm{a}+\mathrm{bi})\times 0=0$
• C. $(\mathrm{a}+\mathrm{bi})\times (\mathrm{c}+\mathrm{di})=(\mathrm{c}+\mathrm{di})\times (\mathrm{a}+\mathrm{bi})$
• D. $(\mathrm{a}+\mathrm{bi})\times 1=(\mathrm{a}+\mathrm{bi})$

Answer 1

The correct option is D

$(\mathrm{a}+\mathrm{bi})\times 1=(\mathrm{a}+\mathrm{bi})$

Step-1: Explanation for correct answer:

For option (D): $(\mathrm{a}+\mathrm{bi})\times 1=(\mathrm{a}+\mathrm{bi})$

We know that the identity property of multiplication states that when a number $\mathrm{A}$ is multiplied by $1$the product is the number itself.

i.e., $\mathrm{A}\times 1=\mathrm{A}$

In option $\left(D\right)$, $\mathrm{A}=\left(\mathrm{a}+\mathrm{bi}\right)$

Therefore, the identity property of multiplication is represented by $(\mathrm{a}+\mathrm{bi})\times 1=(\mathrm{a}+\mathrm{bi})$

Step-2: Explanation for incorrect answer.

For option (A):$(\mathrm{a}+\mathrm{bi})\times \mathrm{c}=(\mathrm{ac}+\mathrm{bci})$

As, $(\mathrm{a}+\mathrm{bi})\ne (\mathrm{ac}+\mathrm{bci})$

Thus, it is not of the form $\mathrm{A}\times 1=\mathrm{A}$ [ The identity property of multiplication ]

For option (B):$(\mathrm{a}+\mathrm{bi})\times 0=0$

As, $(\mathrm{a}+\mathrm{bi})\ne 0$ and $1\ne 0$

Thus, it is not of the form $\mathrm{A}\times 1=\mathrm{A}$ [ The identity property of multiplication ]

For option (C):$(\mathrm{a}+\mathrm{bi})\times (\mathrm{c}+\mathrm{di})=(\mathrm{c}+\mathrm{di})\times (\mathrm{a}+\mathrm{bi})$

As, $(\mathrm{a}+\mathrm{bi})\ne 1$ and $(\mathrm{c}+\mathrm{di})\ne 1$

Thus, it is not of the form $\mathrm{A}\times 1=\mathrm{A}$ [ The identity property of multiplication ]

So, all these options does not verify the identity property of multiplication.

Hence, the correct option is $\left(D\right)$.