### Instruction: What non-zero, real value of $x$ satisfies $(5x)^4= (10x)^3$? Express your answer as a common fraction.### Response: 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 $\frac{(5x)^4}{(5x)^3} = \frac{8 \cdot (5x)^3}{(5x)^3}$.
I can simplify the fractions by canceling out the common factors of $(5x)^3$.
This gives me $(5x) = 8$.
To get $x$ by itself, I can divide both sides by $5$.
This gives me $x = \frac{8}{5}$.