Power towers

I know it sounds trivial, but what is Equal to? it's confusing.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

$a^{b^{c}}=a^{(b^{c})} \neq (a^{b})^{c}$

3Blue1Brown has a great high school lecture on power towers and tetration. Search it up on YouTube.

Ved Pradhan - 11 months, 1 week ago

this is the link.

Zakir Husain - 11 months, 1 week ago

thank you!! :)

I Love Brilliant - 11 months, 1 week ago

See How are exponent towers evaluated?

Pi Han Goh - 11 months, 1 week ago

The second one is correct. The difference between $a^{b^c}$ and $(a^b)^c=a^{bc}$ can be understood in the way:

Suppose you know what is $a^b$ and let it's value be $x$ . Then $a^{bc}=(a^b)^c=x^c$

But you can't evaluate $a^{b^c}$ unless you know value of $a,b,c$ , or you know $b^c=x$ and $a$ thereby finding $a^{b^c}=a^x$

Zakir Husain - 11 months, 1 week ago

thank you for answering! my textbook apparently gets it wrong :\

I Love Brilliant - 11 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$