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Question: How do you simplify \(\left( {3 + 2i} \right)\left( {3 - 2i} \right)\) ?...

How do you simplify (3+2i)(3−2i)\left( {3 + 2i} \right)\left( {3 - 2i} \right) ?

Explanation

Solution

Here they have asked to simplify the expression (3+2i)(3−2i)\left( {3 + 2i} \right)\left( {3 - 2i} \right) . Expand the given expression using FOIL method, where FOIL stands for first, outer, inner, last. The formula for expansion using the FOIL method is given by: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd. After expanding, simplify the expression to get the required answer.

Complete step by step answer:
In this question they have asked to simplify the given expression which is (3+2i)(3−2i)\left( {3 + 2i} \right)\left( {3 - 2i} \right) . Expand the given expression using the FOIL method, FOIL stands for first, outer, inner, last. The formula for expansion using the FOIL method is given by: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd.
Now, apply distributive property for the given expression (3+2i)(3−2i)\left( {3 + 2i} \right)\left( {3 - 2i} \right) , we get
⇒3(3−2i)+2i(3−2i)\Rightarrow 3(3 - 2i) + 2i(3 - 2i)
In order to simplify the above expression, we can again apply the distributive property. Therefore we get
⇒3.3+(3.−2i)+(2i.3)+(2i.−2i)\Rightarrow 3.3 + (3. - 2i) + (2i.3) + (2i. - 2i)
Now, perform multiplication operation as in the above expression to simplify further, we get
⇒9−6i+6i−4i2\Rightarrow 9 - 6i + 6i - 4{i^2}
As we can see in the above expression we have +6i + 6i and −6i - 6i which are having the same number with opposite signs, therefore which will become zero or which gets canceled. So now, the above expression can be written as
⇒9−4i2\Rightarrow 9 - 4{i^2}
We know that i2=(−1)2=−1{i^2} = {\left( {\sqrt { - 1} } \right)^2} = - 1 . Now substitute this value of i2{i^2} in the above expression for further simplification, we get
⇒9−(4×−1)\Rightarrow 9 - (4 \times - 1)
Therefore, we get
⇒9+4=13\Rightarrow 9 + 4 = 13

Therefore the simplified form of (3+2i)(3−2i)\left( {3 + 2i} \right)\left( {3 - 2i} \right) is 1313.

Note:
Whenever we have this type of problem on simplification, we can make use of the foil method formula directly to expand and simplify the expression to get the required answer. If in general, a+bia + bi is a complex number then a−bia - bi is called as the conjugate number, and the product of these two that is (a+bi)(a−bi)(a + bi)(a - bi)will always be a real number, as the answer for the given expression is also a real number.