Question

Consider the following statements:
\[\begin{align}
  & {{S}_{1}}=-8=2i\times 4i=\sqrt{-4}\times \sqrt{-16} \\
 & {{S}_{2}}:\sqrt{\left( -4 \right)}\times \sqrt{\left( -16 \right)}=\sqrt{\left( -4 \right)\times \left( -16 \right)} \\
 & {{S}_{3}}:\sqrt{\left( -4 \right)\times \left( -16 \right)}=\sqrt{64} \\
 & {{S}_{4}}:\sqrt{64}=8 \\
\end{align}\]
Of these statements the incorrect one is,
(a) \[{{S}_{1}}\] only
(b) \[{{S}_{2}}\] only
(c) \[{{S}_{3}}\] only
(d) None of these

Answer 1

Hint: First consider, \[\sqrt{-4}\times \sqrt{-16}\]. Try to find the answer using the mathematical form \[\sqrt{a}\times \sqrt{b}=\sqrt{ab}\] and by using complex form taking \[\sqrt{-1}=i\]. Thus, compare the 4 statements and find the wrong one.

Complete step-by-step answer:
We have been given 4 statements, from which we need to find which all are correct and which is wrong.
Thus let us first find the value of \[\sqrt{-4}\times \sqrt{-16}\].
We all know that, \[\sqrt{a}\times \sqrt{b}=\sqrt{ab}\].
But this mathematical equation holds true only and only when at least one of them is non – negative.
Here we have been given two negative numbers, which are (-4) and (-16). None of them are non – negative. Thus we can’t apply the rule or mathematical equations.
\[\therefore \sqrt{-4}\times \sqrt{-16}=\sqrt{+64}=\pm 8\], but this won’t give us the proper answer.
We know that, \[\sqrt{-1}=i\].
\[\sqrt{-4}=\sqrt{\left( -1 \right)\times 4}=2i\]
Similarly, \[\sqrt{-16}=\sqrt{\left( -1 \right)\times 16}=4i\].
Thus, \[\sqrt{-4}\times \sqrt{-16}=2i\times 4i\] \[\left\{ \because {{i}^{2}}=-1 \right\}\]
\[\sqrt{-4}\times \sqrt{-16}=8{{i}^{2}}=8\times \left( -1 \right)=-8\]
Thus, \[\sqrt{-4}\times \sqrt{-16}=-8\].
Thus the correct answer for this expression of complex numbers is (-8).
Now let us look into \[{{S}_{1}}:-8=2i\times 4i=\sqrt{-4}\times \sqrt{-16}\], now this is equal to the mathematical equation, \[\sqrt{a}\times \sqrt{b}=\sqrt{ab}\]. Thus \[{{S}_{2}}\] is also correct.
\[{{S}_{3}}:\sqrt{\left( -4 \right)\times \left( -16 \right)}=\sqrt{64}\], which is also a correct statement.
Now let us look into \[{{S}_{4}}:\sqrt{64}=8\], which is wrong.
\[\sqrt{64}=\pm 8\]
Thus out of the 4 statements \[{{S}_{4}}\] is wrong.
\[\therefore \] Option (d) is the correct answer.

Note: The reason why most say the answer is only (-8) is because, \[\sqrt{-1}\] is considered to be i, which isn’t a complete answer. Since every number in the complex plane must have exactly 2 distincting roots. We can say that both 8 and -8 could satisfy the equation.