I’m looking for cases of invalid math operations producing (in spite of it all) correct results (aka “every math teacher’s nightmare”).

One example would be “cancelling” the 6’s in

$$\frac{64}{16}.$$

Another one would be something like

$$\frac{9}{2} – \frac{25}{10} = \frac{9 – 25}{2 – 10} = \frac{-16}{-8} = 2 \;\;.$$

Yet another one would be

$$x^1 – 1^0 = (x – 1)^{(1 – 0)} = x – 1\;\;.$$

Note that I am specifically

notinterested inmathematical fallacies(akaspurious proofs). Such fallacies produce shockingly wrong ends by (seemingly) valid means, whereas what I am looking for all cases where one arrives at valid ends by (shockingly) wrong means.Edit: fixed typo in last example.

**Answer**

I was quite amused when a student produced the following when cancelling a fraction:

$$\frac{x^2-y^2}{x-y}$$

He began by “cancelling” the $x$ and the $y$ on top and bottom, to get:

$$\frac{x-y}{-}$$

and then concluded that “two negatives make a positive”, so the final answer has to be $x+y$.

**Attribution***Source : Link , Question Author : Community , Answer Author : Community*