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## 'Climbing Powers' printed from http://nrich.maths.org/

Why do this problem?
The problem demonstrates that mathematical language needs to be
very precise and it is important to avoid ambiguity.

If the learners are encouraged to do as much as they can without
looking at the hint most of them will discover for themselves that
$2^{3^4}$ is ambiguous because $2^{(3^4)}$ is not the same as
$(2^3)^4$.

The problem gives opportunities for all the learners to have some
success and for them to investigate the powers of $\sqrt 2$ using a
calculator or computer and discover the behaviour of the two
different sequences for themselves.

At the same time the problem provides a challenge for the most able
students to prove the results which calls for an understanding of
functions. One proof uses the logarithm function and the other
mathematical induction.

Possible Approach
After the class has worked on indices this problem gives
reinforcement material for everyone and some extension
possibilities for the most able.

The hint gives the learners a good deal of 'scaffolding' thus
enabling the teacher to set the problem for the learners to tackle
independently without a lot of introduction and help from the
teacher, at least initially.

The problem could be set as homework with the intention of going
over the proofs in class the next lesson and having a whole class
discussion.

Key Questions
What does $a^{b^c}$ mean?

What is the difference between $(a^a)^a $ and $a^{(a^a)}$?

If you repeat (or iterate) the function which maps $a^a$ to
$(a^a)^a $what is the next value? What is this function?

If you repeat (or iterate) the function which maps $a^a$ to
$a^{(a^a)} $ what is the next value? What is this function?

Possible support