Answer
$4$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
(1+i)^2(1-i)^2
,$ use the laws of exponents to express the factors in a single exponent. Then use the special product on multiplying the sum and difference of like terms. Also, use $i^2=-1.$
$\bf{\text{Solution Details:}}$
Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
[(1+i)(1-i)]^2
.\end{array}
Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
[(1)^2-(i)^2]^2
\\\\=
[1-i^2]^2
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
[1-(-1)]^2
\\\\=
[1+1]^2
\\\\=
[2]^2
\\\\=
4
.\end{array}
