This is the kind of shower thought I have. I am envious of those that have real shower thoughts

Sincere question: why do you put that negative -27 in parentheses?

-27^2/3 would come out to a different number, because the exponentiation has a higher precedence than the negation

Presumably to avoid ambiguity as to whether the thing to exponentiate is 27 or -27.

Ah, so without parentheses, it would be:
x = -27^2
x / 3?
That doesn't sound very intuitive though...

yup:

(-27)^2/3 = ((-27) ^ 2) / 3

-27^2/3 = (-(27 ^ 2)) / 3

The reason is that you almost always want the negation to happen after the exponentiation, because it otherwise wouldn't have any effect (at least when the exponent is even).

Thanks! 😃

(-27)^2/3^

3^3^ = 27 => 3 = 27^1/3^ => -3 = (-27)^1/3^ => (-27)^2/3^ = ((-27)^1/3^)^2^ = (-3)^2^ = 9

-27^2/3^ is a different answer and problem than (-27)^2/3^.

(-27)^2^ is (-27)(-27) while -27^2^ is (-27)(27).

I like to think of -27^2 as -(27^2).

You're having some complex thoughts.

That's the common shorthand yes, but its still treated like its there.

The other interpretation is just as valid though.

=(-1)^(2/3) * (3^3)^(2/3) 

= e^(2*i*pi/3) * 9 

= (cos(2pi/3) + i * sin(2pi/3)) * 9

= (-1/2 + i * sqrt(3)/2) * 9