Which of the following contain reciprocal expressions?

Question

I.  x^{\frac{1}{2}}  and  \frac{\sqrt{x}}{x}

II.  \sqrt{x^{-2}}  and x

III. x^3 and  x^{\frac{1}{3}}

A. I only
B. II only
C. I and II only
D. II & III only
E. I & III only

NandishSS · Answer

x^{\frac{1}{2}}  and  \frac{\sqrt{x}}{x}  ==> x^(1/2) & x^(-1/2)

\sqrt{x^{-2}}  and x ==>  \frac{1}{x}  & x

Hence C

LevanKhukhunashvili · Answer

1)  \sqrt{x}  reciprocal is  \frac{1}{\sqrt{x}} = \frac{\sqrt{x}}{x} ; Yes

2)  \sqrt{\frac{1}{x^2}} =1/x reciprocal is x; yes

3) x^3 reciprocal is  \frac{1}{x^3} = x^-3 ; NO

IMO
Ans:  C

IanStewart · Answer

The wording of the question is bad -- it's not clear what the question means. If I were to write "1/2 and 3", for example, as an answer choice, then that answer "contains reciprocal expressions" since 1/2 is the reciprocal of something, and so is 3. The question means to ask "in which of the following are the two expressions reciprocals of each other?"

If two numbers or expressions are reciprocals of each other, then their product is 1. In item I, we have √x and √x/x, which have a product of 1, so these are reciprocals.

In item II, we have √(1/x^2) = 1/√(x^2) = 1/|x| along with x. These are only reciprocals if |x| = x, so they're only reciprocals when x is positive. If x is negative, they are negative reciprocals of each other. So this should not be a right answer here, though I won't be surprised if the source incorrectly thinks it is right.

In item III, we certainly do not have reciprocal expressions except when x = 1 or -1.

So the answer should be I only, or A.

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Which of the following contain reciprocal expressions?

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