Question

How do you simplify \[7i\times 3i\times \left( -8-6i \right)\]?

Answer 1

Hint: To simplify the given complex numbers, we need to multiply the imaginary numbers which are present outside the brackets with the imaginary number with complex numbers which are present inside the brackets. Then, we will get the answer to the given question.While solving we will use the value ${{i}^{2}}=-1$ of the complex number for the given problem.

Complete step-by-step solution:
According to the question, we have been asked to simplify the given complex number \[7i\times 3i\times \left( -8-6i \right)\].
To solve this question, we will start with multiplying the imaginary numbers outside the bracket.
For simplification, we will multiply the numbers and $i$ separately. That is, we have to multiply 7 with 3 and $i$ with $i$. So, we can get 21 for the multiplication of 7 & 3. In the same way, we get ${{i}^{2}}$ for the multiplication of $i$ and $i$. Mathematically, we can represent it as
\[\Rightarrow \left( 7\times 3 \right)\left( i\times i \right)\times \left( -8-6i \right)\]
And on further simplifications, we get
\[\Rightarrow 21{{i}^{2}}\times \left( -8-6i \right)\]
We know that \[{{i}^{2}}=-1\] which is the property of imaginary number. Therefore, we get
\[\Rightarrow 21\left( -1 \right)\times \left( -8-6i \right)\]
\[\Rightarrow -21\times \left( -8-6i \right)\]
Now, we will apply the distributive property \[a\left( b+c \right)=ab+ac\], in the term\[-21\times \left( -8-6i \right)\]. Therefore, we get
\[\Rightarrow \left( \left( -21 \right)\times \left( -8 \right) \right)-\left( \left( -21 \right)\times \left( 6i \right) \right)\]
And we can also write it as
\[\Rightarrow \left( 168 \right)+\left( 126i \right)\]
It is as same as
\[\therefore 168+126i\]
Hence, the answer for the simplification of the complex number \[7i\times 3i\times \left( -8-6i \right)\] is \[168+126i\].

Note: While solving this question, we need to be very careful we might make a calculation mistake and end up with an incorrect answer. Also, we can solve this question by first multiplying terms out of the brackets one by one with the terms inside the brackets.