I've solved an equation in two different ways, and I keep getting two different solutions, what's wrong?

Question

Can you tell me where the mistake is?

Answer 1

In the first method, you made the following mistake:

$$x^4\cdot x^3 = x^7 \color{red}{\neq x^{12}}$$

The following statement is what you meant (not that it helps in any way here):

$$\left(x^4\right)^3 = x^{12}$$

Instead, you have $x^4\cdot x^8 = x^{12}$, which yields

$$\frac{6}{x^4} = \frac{6x^8}{x^{12}} \implies \frac{6x^8-1536}{x^{12}} = 0; \quad x^{12} \neq 0$$

from which you obtain the desired result $x = \pm 2$. Note that you missed this in your second way, hence the solution isn’t complete. Any equation in the form $x^2 = a$ has two solutions: $x = \pm\sqrt{a}$.

Answer 2

Your error is on the left, on the second line: $\dfrac{6}{x^4} \neq \dfrac{6x^3}{x^{12}}$.

Correcting this, that second line should read $$\dfrac{6x^8 - 1536}{x^{12}} = 0,$$ so $6x^8 = 1536$, and $x^8 = 256$, so $x = 2$ (or $x = -2$, which solution you missed in both attempts).

Answer 3

$$\frac{x^n}{x^m}=x^{n-m}\implies\frac{x^{12}}{x^4}=x^8$$