Why is $x^{x^x}=x^{x x}=x^{x^2}=x^{2x}$ wrong?

Question

I was solving an excercise that involved the expression:

$$x^{x^x}$$

I decided to apply property of exponentials ($a^{b^c}=a^{bc}$) And I could arrive at the following:

$$x^{x x}=x^{x^2}=x^{2x}$$

But this is clearly wrong, I want to know why is it incorrectly, since it just seems as normal use of exponent laws

Answer 1

This is just an order of operations issue. When we write something like

$${x^{x^{x}}}$$

we (by convention) mean

$${x^{(x^{(x)})}}$$

Now, the exponent laws says

$${(a^{b})^{c}=a^{bc}}$$

Notice the difference in bracket placement? The order of evaluation is different. Now if instead you had

$${(x^{x})^{x}}$$

Then indeed, this equals ${x^{x\times x}=x^{(x^2)}}$. Notice I cannot apply the rule again because of the placement of brackets. So ${x^{(x^2)}\neq x^{2x}}$

Hope that helps :)

Answer 2

Numeric examples are good to provide a check on which expressions are equal. Consider

$$2^{3^2}.$$

Thinking of this as $2^{(3^2)}$ gives $2^{3^2} = 2^9 = 512$. That's not the same as $2^{3\cdot2} = 2^6 = 64$ (which is the same as $(2^3)^2 = 8^2 = 64$).

So $a^{b^c} = a^{bc}$ is not a valid property of exponents.

Answer 3

The problem is that

$$x^{(x^x)} \ne (x^x)^x.$$

And $x^{x^x}$ without parentheses, by convention, means the former, not the latter. And there's nothing you can do with exponent identities to simplify the former: in begets novelty with each added exponent in the tower.