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

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

• $(x^x)^x$ is not the same as $x^{(x^x)}$
– Sil
Aug 8, 2020 at 20:39
• See also the Wikipedia article Tetration, especially the Properties section. Aug 9, 2020 at 8:17

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 :)

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.

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.