This is part of a series on common misconceptions.
Why some people say it's true: and therefore
Why some people say it's false: Since
The statement is .
represents the area of a by square, or a square composed of by squares. Hence, the area is
We can also take a conversion approach. Here, one of the instances of in the will cancel, but another instance of will remain. So we must apply the conversion factor again:
This can also be seen geometrically. Consider a 1-meter long stick. Because we can mark off units along the stick. can be visualized as a area. As you can see in the image below, this area is filled with square areas. Therefore,
Rebuttal: But when you make to , all you do is . The same for So all you do is .
Reply: while . We have to replace the by in both instances. A simple way of remembering this approach is to compare it to the rule of exponent , and hence .