15
votes
How should I convince a student who thinks they proved $1=-1$
Should we impose that $(a^m)^n=a^{mn}$ only when $a \gt 0$?
Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $(a^m)^n=a^{mn}$ cannot be true ...
- 1,940
9
votes
What is the standard for "simplifying your answer"?
Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that $\sqrt{2}/2$ should be preferred to $1/\sqrt{2}$ struck me as ...
- 5,879
9
votes
How should I convince a student who thinks they proved $1=-1$
Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work.
So please don't accept this answer.
Anyway, the educational answer is to see ...
- 22.1k
6
votes
How should I convince a student who thinks they proved $1=-1$
When I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. ...
- 26.9k
3
votes
What is the standard for "simplifying your answer"?
I would say the canonical answer for what constitutes 'simplified as much as possible' is whatever the exam board says it is.
'Simplify' isn't a mathematical function. It is a pedagogical instruction ...
- 5,890
3
votes
Where to find good exercises for term operations?
It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I ...
2
votes
What is the standard for "simplifying your answer"?
There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically. So there is not, and cannot, be a uniquely defined "simplest form" for a given ...
- 26k
2
votes
How should I convince a student who thinks they proved $1=-1$
I think the previous answers have focused too much on the details of rational exponents and negative bases. There is a much simpler point about logic that resolves this whole example and that I think ...
- 31
1
vote
How should I convince a student who thinks they proved $1=-1$
(My new and hopefully improved answer)
Should we require that $(a^m)^n=a^{mn}$ only when $a \gt 0$ ?
That might "solve" the problem in some sense, but we do have legitimate cases with negative ...
- 1,244
1
vote
Where to find good exercises for term operations?
Something like this?
$$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$
$$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \...
- 1,327
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