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This question already has an answer here:

I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3. Please explain me the difference between two expressions.

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marked as duplicate by kingW3, JMoravitz, Claude Leibovici, Henrik, user91500 Jun 3 '17 at 9:22

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Your calculator is interpreting it as this:

$${{2^2}^2}^2 = ((2^2)^2)^2 = (4^2)^2 = 16^2 = 256$$

Google is interpreting it as this:

$${{2^2}^2}^2 = 2^{(2^{(2^2)})} = 2^{(2^4)} = 2^{16} = 65536$$

Similarly with the threes.

Technically Google is correct because order of operations says to do exponents first. So when we want to evaluate $2^{\color{red}{2}^{\color{blue}{2^2}}}$, order of operations says to evaluate the $\color{red}{{2^{\color{blue}{2^2}}}}$ first, i.e., evaluate the exponent first. Apply this rule again and it tells us we're supposed to evaluate the $\color{blue}{2^2}$ first, which is $4.$ Therefore $\color{red}{{2^{\color{blue}{2^2}}}} = 2^4 = 16$, and so $2^{\color{red}{2}^{\color{blue}{2^2}}} = 2^{16} = 65536$.

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  • $\begingroup$ What is correct mathematically? $\endgroup$ – shiv garg Jun 2 '17 at 18:05
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    $\begingroup$ @shivgarg, technically Google is correct because order of operations says to do exponents first. Also see my edit in my answer. $\endgroup$ – tilper Jun 2 '17 at 18:08
  • $\begingroup$ "what is correct mathematically?" is also answered in greater detail in the linked question that you should see a link to above, but here it is again in case you missed it. $\endgroup$ – JMoravitz Jun 2 '17 at 18:09
  • $\begingroup$ In cases like this it never hurts to be explicit about the order you intend. Use parens to instruct Google (or a more modern calculator) how you want it processed. $\endgroup$ – SDsolar Jun 2 '17 at 18:13
  • $\begingroup$ Actually, I don't think there is a "correct" answer. It's a matter of convention, and the convention is not set in stone. $\endgroup$ – Robert Israel Jun 2 '17 at 18:14
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The difference lies in the applied order of operations.

The Casio calculator does operations strictly left to right unless you break it with parentheses:

2^2^2^2 = 4^2^2 = 16^2 = 256

But Google evaluates the entire expression (correctly) using right-to-left order or operations for the stacked powers:

2^2^2^2 = 2^2^4 = 2^16 = 65536

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  • $\begingroup$ What is correct mathematically? $\endgroup$ – shiv garg Jun 2 '17 at 18:06
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    $\begingroup$ I stated that in my answer. $\endgroup$ – John Jun 2 '17 at 18:08

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