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Question: How do you simplify \(\left( {1 - 4i} \right)\left( {2 + i} \right)\) ?...

How do you simplify (1−4i)(2+i)\left( {1 - 4i} \right)\left( {2 + i} \right) ?

Explanation

Solution

In this problem, we have been given a product of two complex numbers and we are asked to simplify the given product of two complex numbers. If we simplify the given terms then the resulting term may also be a complex number. That is the resulting term will be an addition of two terms in which one term will be multiplied by a complex number. Then we will put the necessary values.

Formula used: i=−1i = \sqrt { - 1}
Squaring both the sides,
i2=−1{i^2} = - 1

Complete step by step answer:
Given term is (1−4i)(2+i)\left( {1 - 4i} \right)\left( {2 + i} \right)
Now, we are going to use the distributive property to expand the brackets, that is, we are going to multiply the first term of first bracket with every term of second bracket and second term of first bracket with every term of second bracket.
(1−4i)(2+i)=1(2+i)−4i(2+i)\left( {1 - 4i} \right)\left( {2 + i} \right) = 1\left( {2 + i} \right) - 4i\left( {2 + i} \right)
Opening the brackets,
(1−4i)(2+i)=(1×2)+(1×i)−(4i×2)−(4i×i)\left( {1 - 4i} \right)\left( {2 + i} \right) = \left( {1 \times 2} \right) + \left( {1 \times i} \right) - \left( {4i \times 2} \right) - \left( {4i \times i} \right)
Next we simplify the above term, we get
(1−4i)(2+i)=2+i−8i−4i2\left( {1 - 4i} \right)\left( {2 + i} \right) = 2 + i - 8i - 4{i^2}, here if we add ii and −8i - 8i then we get, −7i - 7i
Simplifying, we get,
(1−4i)(2+i)=2−7i−4i2\left( {1 - 4i} \right)\left( {2 + i} \right) = 2 - 7i - 4{i^2}
Since in complex analysis, i2=−1{i^2} = - 1 , the expression simplifies to
(1−4i)(2+i)=2−7i−4(−1)\left( {1 - 4i} \right)\left( {2 + i} \right) = 2 - 7i - 4\left( { - 1} \right) , the product of two negative terms becomes positive.
⇒(1−4i)(2+i)=2−7i+4\Rightarrow \left( {1 - 4i} \right)\left( {2 + i} \right) = 2 - 7i + 4
Adding the like terms, we get,
(1−4i)(2+i)=6−7i\left( {1 - 4i} \right)\left( {2 + i} \right) = 6 - 7i
Therefore the resultant term is also a complex term. Since it is in the form of addition of two terms in which one term is multiplied by the complex numberii.

So, the answer for the product of two given complex terms is 6−7i6 - 7i .

Note: Here, we got the term −4i2 - 4{i^2} by multiplying −4i - 4i and ii then we put ii to the power of 11 and we get −4(i1i1) - 4\left( {{i^1}{i^1}} \right) . Then, we applied the power rule axay=ax+y{a^x}{a^y} = {a^{x + y}} to combine exponents. By this way, we got −4i2 - 4{i^2} . And the important thing about this problem is the product of any two complex numbers is again a complex number.