Question

Find the square root of $- i$.

Answer 1

We need to find the square root of $\ - i$ . Square root of a number is a value in which it turns to the original number when it is multiplied by itself. Suppose $a$ is a square root of $b$ then it is represented as $a\ = \ \sqrt{b}$ . For example, $3$ is the square root of $9$ then it is represented as $3 = \sqrt{9}$

Complete answer: Given, $- i$
We need to find $\sqrt{- i}$
Let us assume $\sqrt{- i} = \ a -{ib}$
By squaring on both sides ,
$\left( \sqrt{- i} \right)^{2} = \ \left( a - ib \right)^{2}$
By expanding the formula,
We get ,
$i = \ a^{2} + i^{2}b^{2} – 2iab$
$i = a^{2} – b^{2} – 2iab$
By comparing the imaginary part,
$i = - 2iab$
By simplifying we get,
$2ab = - 1$
$ab = - \dfrac{1}{2}$
By comparing the imaginary part,
$i = - 2iab$
By simplifying we get,
$2ab = - 1$
$ab = - \dfrac{1}{2}$
Also by comparing the real part,
$a^{2} – b^{2} = 0$)
Thus we know that,
$\left( a^{2} + b^{2} \right)^{2} = \left( a^{2} – b^{2} \right)^{2} + 4a^{2}b^{2}$
$\left( a^{2} + b^{2} \right)^{2} = \left( a^{2} – b^{2} \right)^{2} + 4{ab}^2$
By substituting the known values,
We get,
$= 0 + 4\left( - \dfrac{1}{2} \right)^{2}$
$= 4\left( \dfrac{1}{4} \right)$
By dividing,
We get,
$=1$
Therefore,
The possibilities of $a$ = $b =$
$\dfrac{1}{\sqrt{2}}$or$\ - \dfrac{1}{\sqrt{2}}$
So by substituting the values in $\sqrt{- i}$ ,
We get the value of square root of$\ - i$
$\sqrt{- i} = \pm \dfrac{1}{\sqrt{2}} – i\dfrac{1}{\sqrt{2}}$
By taking the common terms outside,
We get,
$\sqrt{- i} = \pm \dfrac{1}{\sqrt{2}}\left( 1 – I \right)$
Thus we got the value.
Final answer :
The value of square root of $- i\$ is $\pm \dfrac{1}{\sqrt{2}}\left( 1 – I \right)$

Note:
Mathematically, the symbol $\sqrt{}$ is known as a Radical sign which is used to represent the square root. It is basically one of the methods to solve the quadratic equation . In order to find the square root, we can use two methods
1.Long division method
2.Factorisation method
It’s easy to memorize the square root values of the numbers $1 to 25$. After that number we need to use the method to find the values.