Question

How do you simplify \[\left( 3+4i \right)\left( 3-4i \right)\]?

Answer 1

Hint: Assume the value of the given expression as ‘E’. Use the algebraic identity given as: - \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\], by considering a = 3 and b = 4i, to simplify the given expression. Finally, use the relation: - \[i=\sqrt{-1}\] and \[{{i}^{2}}=-1\], where ‘i’ is the imaginary number, for further simplification of ‘E’ and get the answer.

Complete step by step answer:
Here, we have been provided with the expression \[\left( 3+4i \right)\left( 3-4i \right)\] and we have been asked to simplify it.
Now, let us assume the given expression as ‘E’. So, we have,
\[\Rightarrow E=\left( 3+4i \right)\left( 3-4i \right)\]
Using the algebraic identity: \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\] by considering a = 3 and b = 4i, we get,
\[\Rightarrow E={{3}^{2}}-{{\left( 4i \right)}^{2}}\]
Applying the formula: - \[{{\left( m\times n \right)}^{k}}={{m}^{k}}\times {{n}^{k}}\], we get,
\[\Rightarrow E=9-{{4}^{2}}\times {{i}^{2}}\]
\[\Rightarrow E=9-16{{i}^{2}}\] - (1)
Now, here we can see that in the above expression we have an alphabet ‘i’, actually, it is the notation for the imaginary number \[\sqrt{-1}\]. ‘i’ is the solution of the quadratic equation \[{{x}^{2}}+1=0\]. There are no real solutions of this quadratic equation and therefore the concept of imaginary numbers and complex numbers arises. A complex number is written in a general form as: - \[z=a+ib\], where ‘z’ is the notation of complex numbers, ‘a’ is the real part and ‘ib’ is the imaginary part. Here, \[i=\sqrt{-1}\].
Now, let us come back to the expression ‘E’. Since, \[i=\sqrt{-1}\], therefore on squaring both the sides, we get,
\[\Rightarrow {{i}^{2}}=-1\]s
So, substituting the value of \[{{i}^{2}}\] in equation (1), we get,
\[\begin{align}
  & \Rightarrow E=9-16\times \left( -1 \right) \\
 & \Rightarrow E=9+16 \\
 & \Rightarrow E=25 \\
\end{align}\]
Hence, the above obtained value is the simplified form and our answer.

Note:
One must not consider ‘i’ as any variable or just an alphabet. Remember that ‘i’ always denotes the imaginary number \[\sqrt{-1}\] in the topic ‘complex numbers’. You must remember the formulas of the topic ‘exponents and power’ like: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m-n}},{{a}^{m}}\div {{a}^{n}}{{=}^{m-n}},{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], because these formulas are frequently used in other topics of mathematics. Remember the concepts of complex numbers and their general forms.