Question

Find the value of \[\sqrt 2 \sqrt { - 3} \].
A. \[\sqrt 6 i\]
B. - \[\sqrt 6 i\]
C. \[\sqrt 3 i\]
D. \[\sqrt 6 \]

Answer 1

Hint: In this question, we have to determine the value of \[\sqrt 2 \sqrt { - 3} \]. For this, we have to use the concept of imaginary unit. That is “iota”. The square root of a negative number \[\sqrt { - 3} \] is the primary square root of the positive radicand multiplied by i. After simplification, we get the desired result.

Formula used: Consider the following useful identity of iota for solving this example..
 \[{i^2} = - 1\]
\[i = \sqrt { - 1} \]
Here, i is the imaginary unit called ‘iota’.

Complete step-by-step answer:
Consider an expression \[\sqrt 2 \sqrt { - 3} \]
Now, let us simplify the above expression.
\[\sqrt 2 \sqrt { - 3} = \sqrt 2 \sqrt {3 \times \left( { - 1} \right)} \]
\[\sqrt 2 \sqrt { - 3} = \sqrt 2 \sqrt 3 \sqrt { - 1} \]
But we know that \[{i^2} = - 1 \Rightarrow i = \sqrt { - 1} \]
\[\sqrt 2 \sqrt { - 3} = \sqrt 2 \sqrt {3 \times {i^2}} \]
By simplifying further, we get
\[\sqrt 2 \sqrt { - 3} = \left( {\sqrt {2 \times 3} } \right)i\]
\[\sqrt 2 \sqrt { - 3} = \left( i{\sqrt 6 } \right)\]
Hence, the value of an expression \[\sqrt 2 \sqrt { - 3} \] is \[\left( i{\sqrt 6 } \right)\] .
Therefore, the correct option is \[\left( i{\sqrt 6 } \right)\].

Additional Information: Iota is a Greek letter used in mathematics to represent the imaginary component of a complex number. The all-encompassing set of Complex Numbers was produced once the imaginary I was linked with the collection of Real Numbers. Since there is no real number whose square is a negative integer, imaginary numbers cannot be expressed by a real number. To solve this difficulty, the letter i is invented to signify \[\sqrt { - 1} \].

Note: Many students make mistakes in the simplification part. Many times students write iota, that is i inside the square root sign while finding the square root of a negative number instead of writing it outside the square root sign. This is the only way through which we can solve this example in an easy manner.