# The variable z is often used to denote a complex number and z^- is used to denote its conjugate. If z = a+bi, simplify the expression

The variable z is often used to denote a complex number and z¯ is used to denote its conjugate. If z = a+bi, simplify the expression
${z}^{2}-{\stackrel{―}{z}}^{2}$
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Anonym

With z=a+bi, the conjugate, $\stackrel{―}{z}$, is a-bi. Hence, the expression ${z}^{2}-{\stackrel{―}{z}}^{2}$, is equivalent to
${\left(a+bi\right)}^{2}-{\left(a-bi\right)}^{2}=\left[{\left(a\right)}^{2}+2\left(a\right){\left(bi\right)}^{2}\right]-\left[{\left(a\right)}^{2}-2\left(a\right)\left(bi\right)+{\left(bi\right)}^{2}\right]$
$\left(use{\left(a+b\right)}^{2}={\left(a\right)}^{2}+2\left(a\right)\left(b\right)+{\left(b\right)}^{2}\right)$
$=\left[{a}^{2}+2abi+{b}^{2}{i}^{2}\right]-\left[{a}^{2}-2abi+{b}^{2}{i}^{2}\right]={a}^{2}+2abi+{b}^{2}{i}^{2}-{a}^{2}+2abi-{b}^{2}{i}^{2}=\left({a}^{2}-{a}^{2}\right)+\left(2abi+2abi\right)+\left({b}^{2}{i}^{2}-{b}^{2}{i}^{2}\right)=0+4abi+0=\left(4abi\right)i$.
Hence, ${z}^{2}-{\stackrel{―}{z}}^{2}=\left(4abi\right)i$.