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What does the following statement mean ?

“The expression $((a^b)^c)^d$ has five interpretations. The simplest of all is $a^{bcd}$”

BAI
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Aditi
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  • @BAI thank you very much for editing :) – Aditi Nov 19 '17 at 13:44
  • @bai: Introducing parentheses in your latter edit actually destroys the meaning of the problem, which lies in counting the significant ways to parenthsize the expression (exponentiation not being associative). – hardmath Nov 19 '17 at 16:52

1 Answers1

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I assume the current form of the question is what was intented, and something you might write in unformatted text as a^b^c^d.

The problem is that exponentiation is a nonassociative operation, and the formula doesn't explicitly indicate the order of operations.

There are five different ways you can perform a chain of three consecutive operations (for notational convenience, I will use $\star$ to denote an arbitrary binary operation):

  • $((a \star b) \star c) \star d$
  • $(a \star (b \star c)) \star d$
  • $(a \star b) \star (c \star d)$
  • $a \star ((b \star c) \star d)$
  • $a \star (b \star (c \star d))$

For a nice, associative operation like addition, all five results give the same value. For nonassociative operations, we need to decide which of these five to use (or insist that we never write such ambiguous formulas).

For exponentiation, the standard convention is the last of these; you perform the operations from right to left (top-right to bottom-left in the traditional typesetting). This is called being "right associative".

That is, you should read it as

$$ \huge{a^{\left( b^{\left( c^d \right)} \right)} }$$

  • Thank you. So do you mean that it will have only one interpretation since we can perform the operation in one way only ? – Aditi Nov 19 '17 at 13:55
  • The point is that without parentheses the expression is ambiguous strictly speaking, but there is a convention for which interpretation was intended without parentheses. It is a bit subjective which interpretation is simplest. – hardmath Nov 19 '17 at 16:58