Even roots and exponents power rule

Question

I've a question about exponents power rule applied to even roots.

Assuming as facts

1. $x \in \mathbb{R}$
2. $\sqrt[2n]{x^{2n}} = |x|$
3. $\sqrt[a]{x^b} = x^{b/a}$
4. the exponents power rule $({x^a})^b = x^{a b}$ is true
5. $x^{a b} = x^{b a}$

Problem 1

We have that $\sqrt[2n]{x^{2n}} = {x^{2n}}^{\frac{1}{2n}} = x^1 = x$ applying 3.

How is it possible that $\sqrt[2n]{x^{2n}} = x$ if 2 is true? That means that $|x| = x$ which is clearly not true.

Problem 2

$x^{a/b} = x^{a\frac{1}{b}} = x^{\frac{1}{b}a}$ applying 4 and 5.

But this is not true for all a and b. For instance, if $a=2, b=4$:

1. $x^{2/4} = x^{2\frac{1}{4}} = \sqrt[4]{x^2}$
2. $x^{2/4} = x^{2\frac{1}{4}} = x^{\frac{1}{4}2} = ({\sqrt[4]x})^2$

That results in saying that $ \sqrt[4]{x^2} = ({\sqrt[4]x})^2$ which is false.

Answer 1

When you divide the exponent in $x^{2n/2n}$ you have to remember that, by definition, the square root of any number is always positive, which means that you have to get the module of the expression before doing the actual division

As for the second one, when you go from $\sqrt[4]{x^2}$ to $(\sqrt[4]{x})^2$ you are restricting the domain (from R to R+), which means the two are not equivalent. In order to actually make them equivalent, you have to take the absolute value of x before moving the exponents around