1. **Problem i.a:** Simplify $\left(3x^5\right)^4$ using the laws of indices. 2. The law of indices states that $(ab)^n = a^n b^n$ and $(x^m)^n = x^{mn}$. 3. Apply the law: $\left(3x^5\right)^4 = 3^4 \times \left(x^5\right)^4$. 4. Calculate powers: $3^4 = 81$ and $\left(x^5\right)^4 = x^{5 \times 4} = x^{20}$. 5. So, $\left(3x^5\right)^4 = 81x^{20}$. 6. **Problem i.b:** Simplify $47^{-3}$. 7. The law of indices for negative powers: $a^{-n} = \frac{1}{a^n}$. 8. So, $47^{-3} = \frac{1}{47^3}$. 9. **Problem ii:** Rewrite $\sqrt{9x}^{75}$ as an exponential expression. 10. Recall that $\sqrt{a} = a^{\frac{1}{2}}$. 11. So, $\sqrt{9x} = (9x)^{\frac{1}{2}}$. 12. Raising to the 75th power: $\left((9x)^{\frac{1}{2}}\right)^{75} = (9x)^{\frac{1}{2} \times 75} = (9x)^{\frac{75}{2}}$. 13. Final exponential form: $(9x)^{\frac{75}{2}}$.