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.