Question

How do you simplify \[{{\left( 8+7i \right)}^{2}}\] ?

Answer 1

Hint: To simplify the given problem initially we should find the square the complex number inside the bracket using the algebraic expression. Then solve the expanded terms and reduce it by applying the identities of complex numbers from the basic knowledge of complex numbers to get the final answer as a complex number. Finally represent the complex number in the standard form of representation of a complex number it is \[a\]\[+\]\[ib\].

Complete step by step solution:
The given problem can be simplified as follows,
\[{{\left( 8+7i \right)}^{2}}\]
We know the algebraic identity,
\[{{\left( a+b \right)}^{2}}\]\[=\] \[{{a}^{2}}+2ab+{{b}^{2}}\]
From this, we can write the given equation as below,
\[\Rightarrow \]\[{{\left( 8+7i \right)}^{2}}\] \[=\] \[{{\left( 8 \right)}^{2}}\]\[+\] 2 \[\times \]8\[\times \]7\[i\] \[+\] \[{{\left( 7i \right)}^{2}}\]
Here the basic square of 8 and \[\left( 7i \right)\] are as below,
 \[{{\left( 8 \right)}^{2}}\] \[=\] 8 \[\times \] 8 \[=\] 64
\[{{\left( 7i \right)}^{2}}\] = 7\[i\]\[\times \]7\[i\] \[=\] 49 \[\times \] \[{{i}^{2}}\] = 49 \[\times \] \[{{\left( \sqrt{-1} \right)}^{2}}\] \[=\] 49 \[\times \]\[\left( -1 \right)\] \[=\] \[-\]49
Therefore, substituting them, we get
\[\Rightarrow \] 64 \[+\] 112\[i\] \[+\] \[\left( -49 \right)\]
\[\Rightarrow \] 64 \[-\] 49 \[+\] 112\[i\]
\[\Rightarrow \] 15 \[+\] 112\[i\]
So, the simplified form of the given equation \[{{\left( 8+7i \right)}^{2}}\] will be 15 \[+\] 112\[i\]

Note: This problem can also be solved as a multiple of two identical complex numbers. In which we multiply each part (real and imaginary parts) of one complex number to each parts of another complex number. The knowledge of algebraic identities should be known for solving this problem in this method and even the knowledge of finding the squaring of numbers is necessary. Finally the complex number obtained by simplifying the problem should be necessarily kept in standard form of the representation of a complex number.