Question

How do you simplify: $\left( 7-6i \right)\left( -8+3i \right)$?

Answer 1

Assume the value of the given expression as ‘E’. Multiply each term of the first expression (7 – 6i) with each term of the second expression (-8 + 3i). 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 solution:
Here, we have been provided with the expression $\left( 7-6i \right)\left( -8+3i \right)$ and we have been asked to simplify it.
Now, let us assume the given expression as ‘E’. So, we have,
$\Rightarrow E=\left( 7-6i \right)\left( -8+3i \right)$
Here we have to multiply each term of the first expression (7 – 6i) with each term of the second expression (-8 +3i) to get the answer. So, we have,

& \Rightarrow E=-56+21i+48i-18{{i}^{2}} \\\ & \Rightarrow E=-56+69i-18{{i}^{2}} \\\ \end{aligned}$$ Now, here we can see that in the above expression we have an alphabet ‘i’, actually it is the notation for the imaginary number $$\sqrt{-1}$$. ‘i’ is the solution of the quadratic equation $${{x}^{2}}+1=0$$. There are no real solutions of this quadratic equation and therefore the concept of imaginary numbers and complex numbers arises. A complex number is written in general form as: - $$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$$ So, substituting the value of $${{i}^{2}}$$ in expression ‘E’, we get, $$\begin{aligned} & \Rightarrow E=-56+69i-18\times \left( -1 \right) \\\ & \Rightarrow E=-56+69i+18 \\\ & \Rightarrow E=-38+69i \\\ \end{aligned}$$ Hence, the above obtained value is the simplified form and our answer. **Note:** One must not consider ‘i’ as any variable or just an alphabet. Remember that ‘i’ always denotes the imaginary number $$\sqrt{-1}$$ in the topic ‘complex numbers’. You must remember certain algebraic identities and the formulas of the topic ‘exponents and power’ like: - $${{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 the topic ‘complex numbers’. Remember the concepts of complex numbers and their general forms.