Question

If $z=1+2i$, then find the value of ${{z}^{2}}$.

Answer 1

Hint: To evaluate the value of the given algebraic expression, substitute the value of the given complex number in the given expression, and simplify it using the algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$. Calculate the value of the expression using the fact that $i$ is a root of unity.

Complete step-by-step solution -
We know that $z=1+2i$. We have to calculate the value of ${{z}^{2}}$.
 We observe that $z=1+2i$ is a complex number while ${{z}^{2}}$ is an algebraic expression. We will evaluate the value of this expression at $z=1+2i$.
To do so, we will substitute $z=1+2i$ in the given expression and simplify it using the algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$.
Thus, we have ${{z}^{2}}={{\left( 1+2i \right)}^{2}}$.
Simplifying the above expression using the algebraic identity, we have ${{z}^{2}}={{\left( 1+2i \right)}^{2}}={{1}^{2}}+{{\left( 2i \right)}^{2}}+2\left( 2i \right)\left( 1 \right)$
So, we have \[{{z}^{2}}={{\left( 1+2i \right)}^{2}}={{1}^{2}}+{{\left( 2i \right)}^{2}}+2\left( 1 \right)\left( 2i \right)=1+4{{i}^{2}}+4i\].
We know that $i=\sqrt{-1}$. Thus, we have ${{i}^{2}}=-1$.
Simplifying the above expression, we have \[{{z}^{2}}={{\left( 1+2i \right)}^{2}}={{1}^{2}}+{{\left( 2i \right)}^{2}}+2\left( 1 \right)\left( 2i \right)=1+4{{i}^{2}}+4i=1+4\left( -1 \right)+4i\].
We can further solve the above expression to write it as \[{{z}^{2}}={{\left( 1+2i \right)}^{2}}={{1}^{2}}+{{\left( 2i \right)}^{2}}+2\left( 1 \right)\left( 2i \right)=1+4{{i}^{2}}+4i=1+4\left( -1 \right)+4i=1-4+4i=-3+4i\].
Hence, the value of ${{z}^{2}}$ when $z=1+2i$ is \[-3+4i\].

Note: We must keep in mind that $i=\sqrt{-1}$ is the root of unity. It is a solution to the equation ${{x}^{2}}+1=0$. Thus, we have ${{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1$. We can write any complex number in the form $a+ib$, where $ib$ is the imaginary part and $a$ is the real part. We can’t solve this question without using the algebraic identity.