Question

If a > 0 and b < 0, then \[\sqrt{a}\sqrt{b}\] is equal to (where, \[i=\sqrt{-1}\])
(a) \[-\sqrt{a.\left| b \right|}\]
(b) \[\sqrt{a.\left| b \right|}i\]
(c) \[\sqrt{a.\left| b \right|}\]
(d) None of these

Answer 1

Hint: Here b < 0, which means that b has negative value. Thus find the value of \[\sqrt{b}\] where b is negative, so find \[\sqrt{-b}\]. Now a > 0, a has positive values so \[\sqrt{a}\]. Now find \[\sqrt{a}\sqrt{b}\] and simplify it.

Complete step-by-step answer:
In this question we have been given two conditions a > 0 and b < 0.
Hence we need to find \[\sqrt{a}\sqrt{b}\].
Now b < 0, which means that the values are negative, hence we can write \[\sqrt{b}\] as \[\sqrt{\left| b \right|.i}\].
i.e. \[\sqrt{\left( -b \right)}=\sqrt{\left| b \right|\left( -1 \right)}=i\sqrt{\left| b \right|}\], this is formed because b is negative.
We know that a > 0. Hence, \[\sqrt{a}\] can be written as such. Thus, we can write \[\sqrt{a}\sqrt{b}\] as,
\[\sqrt{a}\sqrt{b}=\sqrt{a}\sqrt{\left| b \right|}.i\]
We got, \[\sqrt{b}=\sqrt{\left| b \right|}.i\]
Thus simplifying the above expression we get,
\[\sqrt{a}\sqrt{b}=\sqrt{a\left| b \right|}.i\]
Thus we got the value of \[\sqrt{a}\sqrt{b}\] as \[\sqrt{a\left| b \right|}.i\].
\[\therefore \] Option (b) is the correct answer.

Note: From the given condition a > 0 and b < 0, you should be able to understand the fact that a signifies positive numbers and b signifies negative numbers. Thus \[\sqrt{a}\] will be as such, we need to find the value of \[\sqrt{-b}\], as b is any number, which is negative.