Question: User: $(x - 3)^{4/3} = 625$
1. **State the problem:** Solve the equation $$ (x - 3)^{\frac{4}{3}} = 625 $$ for $x$.
2. **Recall the formula and rules:** To solve an equation with a fractional exponent, we can raise both sides to the reciprocal power to isolate the base. Here, the exponent is $\frac{4}{3}$, so the reciprocal is $\frac{3}{4}$.
3. **Apply the reciprocal power:**
$$ \left((x - 3)^{\frac{4}{3}}\right)^{\frac{3}{4}} = 625^{\frac{3}{4}} $$
4. **Simplify the left side:**
$$ (x - 3)^{\cancel{\frac{4}{3} \times \frac{3}{4}}} = (x - 3)^1 = x - 3 $$
5. **Calculate the right side:**
First, rewrite $625$ as a power of $5$:
$$ 625 = 5^4 $$
Then,
$$ 625^{\frac{3}{4}} = (5^4)^{\frac{3}{4}} = 5^{4 \times \frac{3}{4}} = 5^3 = 125 $$
6. **Solve for $x$:**
$$ x - 3 = 125 $$
$$ x = 125 + 3 = 128 $$
7. **Final answer:**
$$ \boxed{128} $$