Exponents help!

[Chillupon]

If you had to put the meaning of "^" into words, what would you describe it as? I understand it as 'times something by itself x many times'

2 ^ 2 = 4. Understood. 2 multiplied by itself 2 times. Simple.

2 ^ 2 ^ 2 = 16. Understood. 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. Simple.

2 ^ 2 ^ 2 ^ 2 = 65536. What? 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. 16 multiplied by itself twice, to equal 256, right? No, 65,536. Why?

I don't understand exponents. Can someone in simple terms explain to me why this happens? Why it jumps up to 65,536 so suddenly? This is really confusing me. What am I doing wrong here?

Thanks.

 

[stapel]

2 ^ 2 ^ 2 = 16. Understood. 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. Simple.

But it's "understood" only because you're working with 2:

. . . . .\(\displaystyle (2^2)^2\, =\, (4)^2\, =\, 16\)

. . . . .\(\displaystyle 2^{(2^2)}\, =\, 2^{(4)}\, =\, 16\)

But if you're working with 3, you get:

. . . . .\(\displaystyle (3^3)^3\, =\, (27)^3\, =\, 19,683\)

. . . . .\(\displaystyle 3^{(3^3)}\, =\, 3^{(27)}\, =\, 7.635597485\, \times\, 10^{12}\)

So grouping (that is, stating clearly exactly which power is on exactly what base) is crucial to the "meaning" of the "carat" symbol and to how the exponentiation will be interpreted.

I suspect that you understand "exponents" okay; it's maybe the ambiguity of certain formatting (especially online) that's causing issues...?

 

[soroban]

Hello, Chillupon!


We know that: \(\displaystyle 3^2 \:=\:3\cdot3 \:=\:9\)

But if we are given an "exponential stack",
. . we must read from the top.


Given: \(\displaystyle 3^{3^2}\)

We start with the "uppermost" exponent:
/. \(\displaystyle 3^{(3^2)} \;=\;3^9 \;=\;19,683\)


Given: \(\displaystyle 3^{2^{2^2}} \)

We have:

. . \(\displaystyle 3^{2^{(2^2)}} \;=\;3^{(2^4)} \;=\;3^{16} \;=\;43,\!046,\!721\)


If we are to start from the bottom,
. . parentheses are used.

. . \(\displaystyle {\color{red}(}3^3{\color{red})}^{^2} \;=\;27^2 \;=\;729\)