### Video Transcript

If π₯ is equal to root four over
root two, which of the following is equal to π₯ to the power of negative one? (A) One, (B) root two, (C) negative
root two, (D) root two over two, or (E) negative root two over two.

The value π₯ is the quotient of two
square roots. Before we consider the value of π₯
to the power of negative one, we can simplify the value of π₯ itself. Four is a square number, and its
square root is equal to two. Hence, the value of π₯ is equal to
two over root two. Now, this expression involves a
radical in the denominator. And we may be concerned that we
need to rationalize this denominator before we can proceed. But letβs first consider the
expression we are asked to evaluate. Itβs π₯ to the power of negative
one.

We can recall the law for negative
exponents, which states that for any nonzero real value of π, π to the power of
negative π is equal to one over π to the πth power. Hence, π₯ to the power of negative
one is equal to one over π₯ to the first power, which is simply one over π₯. So what weβre actually being asked
to find is the reciprocal of π₯. This means that the radical in the
denominator of π₯ will be in the numerator of its reciprocal. So we donβt need to worry about
rationalizing the denominator of π₯ before we can continue.

Substituting our simplified
expression for π₯ gives one over π₯ is equal to one divided by two over root
two. Dividing by a fraction is
equivalent to multiplying by the reciprocal of that fraction. So one over π₯ is equal to one
multiplied by root two over two, which is just equal to root two over two. We could also have got to this
point by simply swapping the numerator and denominator of π₯ around to find its
reciprocal. Looking at the five options given,
the correct answer is option (D).

Now, if we had decided to
rationalize the denominator in our expression for π₯ by multiplying both the
numerator and denominator by root two, we wouldβve obtained π₯ is equal to root
two. Then, when evaluating one over π₯,
we wouldβve obtained one over root two. And the denominator wouldβve
required rationalizing again. Multiplying both the numerator and
denominator by root two again would give root two over two, which is the same as our
previous answer. So both methods give the same
result.

Hence, by recalling the law for
negative exponents and applying this to two equivalent expressions for π₯, weβve
found that if π₯ is equal to root four over root two, the value of π₯ to the power
of negative one is root two over two.