1. **Problem statement:** Simplify the expressions using the laws of indices. 2. **Recall the laws of indices:** - $(a^m)^n = a^{m \times n}$ - $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ 3. **Part a) Simplify $(3x^5)^4$:** - Apply the power to both the coefficient and the variable: $(3)^4 \times (x^5)^4$ - Use the law $(a^m)^n = a^{m \times n}$ for the variable: $3^4 \times x^{5 \times 4}$ - Calculate powers: $3^4 = 81$ and $5 \times 4 = 20$ - So, the simplified form is $$81x^{20}$$ 4. **Part b) Simplify $\left(\frac{4}{7}\right)^{-3}$:** - Use the negative exponent rule: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ - So, $\left(\frac{4}{7}\right)^{-3} = \left(\frac{7}{4}\right)^3$ - Calculate the cube: $7^3 = 343$ and $4^3 = 64$ - So, the simplified form is $$\frac{343}{64}$$