Question: following fraction in simplest form, only using positive exp
$\frac{(3t^{-1})^{-3}}{4t^{7}j}$
1. **State the problem:** Simplify the fraction $$\frac{(3t^{-1})^{-3}}{4t^{7}j}$$ so that all exponents are positive.
2. **Recall the exponent rules:**
- Power of a power: $$(a^m)^n = a^{mn}$$
- Negative exponent: $$a^{-m} = \frac{1}{a^m}$$
- When dividing like bases, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$
3. **Simplify the numerator:**
$$ (3t^{-1})^{-3} = 3^{-3} \cdot (t^{-1})^{-3} = 3^{-3} \cdot t^{(-1) \times (-3)} = 3^{-3} \cdot t^{3} $$
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:**
$$ \frac{\frac{t^{3}}{27}}{4t^{7}j} = \frac{t^{3}}{27} \times \frac{1}{4t^{7}j} = \frac{t^{3}}{27 \cdot 4 t^{7} j} = \frac{t^{3}}{108 t^{7} j} $$
6. **Simplify the powers of $t$ in numerator and denominator:**
$$ \frac{t^{3}}{t^{7}} = t^{3-7} = t^{-4} = \frac{1}{t^{4}} $$
7. **Final simplified expression:**
$$ \frac{1}{108 t^{4} j} $$
**Answer:** $$\boxed{\frac{1}{108 t^{4} j}}$$