2022-11-30

Which equation illustrates the identity property of multiplication? A$\left(a+\mathrm{bi}\right)×c=\left(\mathrm{ac}+\mathrm{bci}\right)$ B$\left(a+\mathrm{bi}\right)×0=0$ C$\left(a+\mathrm{bi}\right)×\left(c+\mathrm{di}\right)=\left(c+\mathrm{di}\right)×\left(a+\mathrm{bi}\right)$ D$\left(a+\mathrm{bi}\right)×1=\left(a+\mathrm{bi}\right)$

marystevensbUS

Expert

$\left(a+\mathrm{bi}\right)×1=\left(a+\mathrm{bi}\right)$
Step-1: Explanation for correct answer:
For option (D): $\left(a+\mathrm{bi}\right)×1=\left(a+\mathrm{bi}\right)$
We know that the identity property of multiplication states that when a number $A$ is multiplied by $1$the product is the number itself.
i.e., $A×1=A$
In option $\left(D\right)$, $A=\left(a+\mathrm{bi}\right)$
Therefore, the identity property of multiplication is represented by $\left(a+\mathrm{bi}\right)×1=\left(a+\mathrm{bi}\right)$
Step-2: Explanation for incorrect answer.
For option (A):$\left(a+\mathrm{bi}\right)×c=\left(\mathrm{ac}+\mathrm{bci}\right)$
As, $\left(a+\mathrm{bi}\right)\ne \left(\mathrm{ac}+\mathrm{bci}\right)$
Thus, it is not of the form $A×1=A$ [ The identity property of multiplication ]
For option (B):$\left(a+\mathrm{bi}\right)×0=0$
As, $\left(a+\mathrm{bi}\right)\ne 0$ and $1\ne 0$
Thus, it is not of the form $A×1=A$ [ The identity property of multiplication ]
For option (C):$\left(a+\mathrm{bi}\right)×\left(c+\mathrm{di}\right)=\left(c+\mathrm{di}\right)×\left(a+\mathrm{bi}\right)$
As, $\left(a+\mathrm{bi}\right)\ne 1$ and $\left(c+\mathrm{di}\right)\ne 1$
Thus, it is not of the form $A×1=A$ [ The identity property of multiplication ]
So, all these options does not verify the identity property of multiplication.
Hence, the correct option is $\left(D\right)$.

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