0

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.

Danny
  • 183
  • 3
    $(2^3)^2$ is very different from $2^{3^2}$. –  Mar 27 '18 at 21:18
  • 2
    Also.... this is exactly why parentheses are important to include. –  Mar 27 '18 at 21:18
  • Convention, at least in western society, is going from left to right. I.e. a^b^c would be the same as $(a^b)^c$. – ty. Mar 27 '18 at 21:20
  • @ty. I have never seen anyone follow or propose following that convention (except you if that is what you actually propose). I have always seen it be interpreted from right to left. If it actually were to be interpreted from left to right, then there would be no reason to ever write anything (without parentheses) as a tower of height higher than two since $a^{b^{c^d}}$ could be written much more easily as $a^{bcd}$ – JMoravitz Mar 27 '18 at 21:24
  • OP: You can't use LaTeX because it requires you to use parentheses? But you use parentheses throughout your question! – Brian Tung Mar 27 '18 at 21:25
  • @JMoravitz You are right. This is just my experience with this silly notation. – ty. Mar 27 '18 at 21:30
  • Possible duplicate of What is the order when doing $x^{y^z}$ and why? – Rob Arthan Mar 27 '18 at 22:00

4 Answers4

1

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.

Arthur
  • 199,419
  • Right associativity for exponents (i.e. x^y^z meaning x^(y^z)) is extremely common in any source that actually pays attention to the topic of interpreting ambiguous notation. –  Mar 27 '18 at 21:20
  • Thanks a lot! Remember this lession^_^ – Danny Mar 27 '18 at 21:54
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 $2^{(3^2)}=2^9=512$

JMoravitz
  • 79,518
1

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 $a^r$.

MPW
  • 43,638
0

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

Rob Arthan
  • 48,577