What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? Express your answer as a common fraction.
To solve for $x$, I need to isolate it on one side of the equation. I can start by simplifying the equation a bit, by noting that $10x = 2 \cdot 5x$. So, the equation becomes $(5x)^4 = (2 \cdot 5x)^3$. I can expand the right-hand side using the power of a product rule: $(2 \cdot 5x)^3 = 2^3 \cdot (5x)^3 = 8 \cdot (5x)^3$. Now the equation is $(5x)^4 = 8 \cdot (5x)^3$. I can divide both sides by $(5x)^3$, as long as $x \neq 0$, which it isn't, since the problem says so. This gives me $(5x) = 8$. To get $x$ by itself, I can divide both sides by $5$. This gives me $x = \frac{8}{5}$. This is the value of $x$ that satisfies the equation, and it is already in the form of a common fraction.

