1965 AMC 12 Problem 11

Question

Consider the statements:

\begin{aligned} \mathrm{I:}\quad &(\sqrt{-4})(\sqrt{-16})\\ &=\sqrt{(-4)(-16)}, \end{aligned}

\mathrm{II:}\quad \sqrt{(-4)(-16)}=\sqrt{64},

and

\mathrm{III:}\quad \sqrt{64}=8.

Of these the following are incorrect:

none

\mathrm{I}

only

\mathrm{II}

only

\mathrm{III}

only

\mathrm{I}

and

\mathrm{III}

only

The product rule for square roots requires nonnegative real radicands

Evaluate statements II and III directly

Statement I improperly applies

\sqrt a\sqrt b=\sqrt{ab}

to negative radicands; over the reals its left side is undefined, and with principal complex roots it equals

-8,

not

8.

Statements II and III both correctly simplify to

8.

Thus only I is incorrect.

Therefore, the correct answer is B.

Problem 11 in other years