Question: Express the following fraction in simplest form, only using
$$(3t^{-1})^{-3} \over 4t^{7}j$$
1. **State the problem:** Simplify the expression $$\frac{(3t^{-1})^{-3}}{4t^{7}j}$$ so that it contains only positive exponents.
2. **Recall the exponent rules:**
- $$(a^m)^n = a^{mn}$$
- $$a^{-m} = \frac{1}{a^m}$$
- When dividing, subtract exponents of like bases.
3. **Simplify the numerator:**
$$ (3t^{-1})^{-3} = 3^{-3} \cdot (t^{-1})^{-3} = 3^{-3} \cdot t^{3} $$
4. **Rewrite the numerator with positive exponents:**
$$ 3^{-3} = \frac{1}{3^3} = \frac{1}{27} $$
So numerator becomes:
$$ \frac{t^{3}}{27} $$
5. **Rewrite the entire fraction:**
$$ \frac{\frac{t^{3}}{27}}{4t^{7}j} = \frac{t^{3}}{27} \cdot \frac{1}{4t^{7}j} = \frac{t^{3}}{27 \cdot 4 \cdot t^{7} \cdot j} $$
6. **Combine constants and variables:**
$$ = \frac{t^{3}}{108 t^{7} j} $$
7. **Simplify the powers of $t$ by subtracting exponents:**
$$ \frac{t^{3}}{t^{7}} = t^{3-7} = t^{-4} = \frac{1}{t^{4}} $$
8. **Final simplified expression with positive exponents:**
$$ \frac{1}{108 t^{4} j} $$
**Answer:**
$$ \boxed{\frac{1}{108 t^{4} j}} $$