Question

How do you simplify \[\sqrt{-36}\]?

Answer 1

Hint: This question is from the topic of algebra. For solving this question, first we will know what is the symbol for the square root of negative of 1. After that, we will find the square root of 36 and multiply with that symbol. After that, we will do the further process and solve the question to get the perfect answer.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to simplify the term \[\sqrt{-36}\] or we can say that we have to find the square root of negative 36.
As we know that any number (say, x) multiplied with 1, then the multiplication value will be the same number (that is x).
So, the term \[\sqrt{-36}\] can also be written as \[\sqrt{-1\times 36}\].
So, we can write
\[\sqrt{-36}=\sqrt{-1\times 36}\]
The above equation can also be written as
\[\Rightarrow \sqrt{-36}=\sqrt{-1}\times \sqrt{36}\]
As we know that the square root of negative 1 can also be written as iota (where the symbol of iota is \[i\]). Or, we can say \[i=\sqrt{-1}\]
So, we can write the above equation as
\[\Rightarrow \sqrt{-36}=i\times \sqrt{36}\]
Now, we will write the prime factorization of 36.
Prime factorization of 36 will be
\[36=6\times 6\]
So, we can write the above equation as
\[\Rightarrow \sqrt{-36}=i\times \sqrt{6\times 6}\]
The above equation can also be written as
\[\Rightarrow \sqrt{-36}=i\times \sqrt{{{6}^{2}}}\]
As we know that \[\sqrt{a}\] can be written as \[{{a}^{\dfrac{1}{2}}}\].
So, the above equation can be written as
\[\Rightarrow \sqrt{-36}=i\times {{\left( {{6}^{2}} \right)}^{\dfrac{1}{2}}}\]
As we know that \[{{\left( {{a}^{m}} \right)}^{n}}\] can also be written as \[{{a}^{m\times n}}\]
So, we can write the above equation as
\[\Rightarrow \sqrt{-36}=i\times {{6}^{2\times \dfrac{1}{2}}}\]
The above equation can also be written as
\[\Rightarrow \sqrt{-36}=i\times 6=6i\]
Hence, we get that the simplified value \[\sqrt{-36}\] is \[6i\].

Note:
For solving this type of question, we should have a better knowledge in the topic of algebra. Also, we should know the formulas and properties of the chapter complex number. Always remember that the value of iota ( where the symbol of iota is \[i\]) is square root of 1 that is \[i=\sqrt{-1}\]. Don’t forget the square root of some specific numbers like \[\sqrt{36}=6\] to solve this type of question easily.