This is part of a series on common misconceptions.
True or False?
For all real numbers and
Why some people say it's false:
There are other cases to consider besides positive real numbers. In those other cases, this "identity" may fail.
This statement is . In particular, is true except when and are both negative.
When square roots are fairly straightforward, but keep in mind the operator is a function that gives exactly one value. For example, However, when there is no real number whose square is Thus, when we define where (the imaginary unit). For example,
First, let's check what happens when exactly one of or is negative. For example, if and then Since and , we can multiply these square roots to obtain
Then, since we have so the two sides are equal!
However, we also need to check what happens when both and are negative. In this case,
On the other hand, since and are both negative, Thus,
so the statement does not hold true.
Rebuttal: Do we really need to take the square root before multiplying the radicands?
Reply: Yes! The square root sign is essentially a fractional exponent, and we have to take care of exponents first according to order of operations.
Rebuttal: Why doesn't the "squaring both sides" of the equation argument work?
Reply: Keep in mind that does not imply that While this does not mean that must be equal to
Rebuttal: My teacher told me that
Reply: Well, hopefully your teacher mentioned that this only holds true under certain conditions! In particular, it is true when and are both positive, which is where you usually encounter such equations. It is also true if exactly one of or is negative, but it is false when both are negative, as shown above.
Rebuttal: Why must we say that for Couldn't we set as well? For example,
Reply: No! Remember that for is defined as the non-negative number whose square is Similarly, the convention is that, for is defined as times the non-negative number whose square is
Note that is a function, so it's important that it is defined to have a single output for each input. Many complicated issues (which are tackled in the subject of complex analysis) would arise if we did not use these conventions!
Want to make sure you've got this concept down? Try these problems: