Question

How do you simplify $\left( 6i \right)\left( -2i \right)$?

Answer 1

We have been given a multiplication of complex numbers. We first try to find the sign of the multiplication from the coefficients. Then we multiply the constants and the complex digits separately. We multiply them to find the solution of $\left( 6i \right)\left( -2i \right)$. We also use the identity value of complex numbers where ${{i}^{2}}=-1$.

Complete step by step solution:
We have been given a multiplication of two terms $\left( 6i \right)\left( -2i \right)$.
The terms being $6i$ and $-2i$. One of the terms is positive and the other one is negative.
There is only one negative term which makes the whole multiplication negative.
Therefore, the sign is known to us. We just need to find the multiplication of $6i$ and $2i$.
We multiply all the terms all together.
The terms are constant and complex digits all being in multiplication.
Therefore, $6i\times 2i=6\times i\times 2\times i$.
We multiply the constants first and get $6\times 2=12$.
Then we multiply the variables and get $i\times i={{i}^{2}}$.
The total multiplication becomes $6i\times 2i=6\times i\times 2\times i=12{{i}^{2}}$.
We know that the value ${{i}^{2}}=-1$. Putting the value, we get $12{{i}^{2}}=-12$
Now we have to apply the negative sign for the total simplification.
So, $\left( 6i \right)\left( -2i \right)=-\left( -12 \right)=12$.

The simplified form of $\left( 6i \right)\left( -2i \right)$ is 12.

Note: For any multiplication and to find the appropriate signs for that we can follow the rule where multiplication of same signs gives positive result and multiplication of opposite signs give negative result. We can also multiply all the constants and the variables together to make the process simple.