Question

Consider the following statements:

& {{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{aligned}$$ Of these statements the incorrect one is, (a) $${{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.