Confusion when computing power's power

Question

When I was playing my calculator, I found that 2^2^3 (I cannot use LaTeX since it requires me to use parentheses) and 2^3^2 yield different results. Isn't that by convention, both 2^2^3 and 2^3^2 can be viewed as 2^6 (multiply the two powers first)? However, 2^2^3 results in 256, and 2^3^2 results in 512. So could anyone tell me what should I do to get the right result when there are no parentheses for computing the power's power? Thanks in advance.

Answer 1

Be careful. Writing 2^2^3 can be thought of as ambiguous. It is standard to read power-towers from top down. That is to say, without parentheses, a^b^c should be interpreted as $a^{(b^c)}$, not as $(a^b)^c$

The property you think you are remembering is that $(a^b)^c=a^{b\times c}$, in which case yes, $(2^2)^3=(2^3)^2=2^6=64$

2^2^3 is actually to be interpreted as $2^{(2^3)}=2^8=256$

On the other hand, 2^3^2 is to be interpreted as $2^{(3^2)}=2^9=512$

Answer 2

If you take a calculator and press 2^3^2, whether you get $2^9$ or $2^6$ (or an error) will be entirely up to the manufacturer. There is no convention for this, and to human readers it is entirely ambiguous.

When writing, you should specify either $2^{3^2}$ or $(2^3)^2$, and when using a calculator, you ought to use parentheses and type either 2^(3^2) or (2^3)^2. The same goes for 2^2^3, of course.

Answer 3

In the absence of grouping symbols (parentheses), the expression $a^{b^c}$ means $a^{(b^c)}$, not $(a^b)^c$. Calculators may or may not perform the implicit grouping correctly.

So on a calculator, you should either use parentheses, or store the intermediate result (the exponent $r=b^c$), then compute the final result $a^r$.

Answer 4

Your calculator is correctly implementing the (hotly disputed) correct answers to What is the order when doing $x^{y^z}$ and why?