Question

How do you find the square root of \[ - 63\] ?

Answer 1

Hint: In the above question, we are given a number \[ - 63\] . We have to find the square root of the given number \[ - 63\] . Since, as we know, the square of any real number is always a positive real number. But here, the given number is \[ - 63\] which is a negative integer. That clearly states that \[ - 63\] can not be a square of any real number. Therefore, the square root of \[ - 63\] is not a real number, in fact it is an imaginary i.e. a complex number.

Complete step by step answer:
Given integer is \[ - 63\] .
We have to find the square root of the above given integer.
Since, the integer is negative, hence its square root would be an imaginary number i.e. a complex number.
In doing complex number calculations, we use the number iota which is denoted by the variable \[i\] and is defined as,
\[ \Rightarrow i = \sqrt { - 1} \]
Now our aim is to find the square root of \[ - 63\] i.e. \[\sqrt { - 63} \] .
Now we can also write \[\sqrt { - 63} \] in the form,
\[ \Rightarrow \sqrt { - 63} \]
Or,
\[ \Rightarrow \sqrt {63 \times \left( { - 1} \right)} \]
Separating the square root for both the numbers, we can write
\[ \Rightarrow \sqrt {63} \cdot \sqrt { - 1} \]
Since, \[i = \sqrt { - 1} \]
Therefore,
\[ \Rightarrow i\sqrt {63} \]
After factorising \[63\] , we can write it as \[63 = 9 \times 7 = 3 \times 3 \times 7\]
Therefore, putting the above value in the expression, we get
\[ \Rightarrow i\sqrt {3 \times 3 \times 7} \]
Taking \[3\] out of the root, we have
\[ \Rightarrow 3i\sqrt 7 \]
Hence,
\[ \Rightarrow 3\sqrt 7 i\]
Therefore, the square root of \[ - 63\] is \[3\sqrt 7 i\].

Note:
We can further rewrite the obtained answer in a more efficient way after putting the value of \[\sqrt 7 \] in the obtained square root.
We know that,
\[ \Rightarrow \sqrt 7 = 2.6457\]
Multiplying both sides by \[3\] , we get
\[ \Rightarrow 3\sqrt 7 = 3 \times 2.6457\]
Or,
\[ \Rightarrow 3\sqrt 7 = 7.93725\]
Multiplying both sides by \[i\] , we can write
\[ \Rightarrow 3\sqrt 7 i = 7.93725i\]
Therefore, the square root of \[ - 63\] is \[3\sqrt 7 i\] i.e. \[7.93725i\] .