1. **Problem statement:** Simplify each expression by removing brackets and expressing in simplest form. 2. **Formula and rules:** - Power of a power: $\left(x^m\right)^n = x^{mn}$ - Power of a product: $\left(ab\right)^n = a^n b^n$ - Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ - Negative exponent rule: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ 3. **Step-by-step simplifications:** **a.** $a \left(3a^2\right)^2 = a \times 3^2 \times \left(a^2\right)^2 = a \times 9 \times a^{4} = 9a^{5}$ **b.** $\left(\frac{2}{p^2 q}\right)^4 = \frac{2^4}{\left(p^2\right)^4 q^4} = \frac{16}{p^{8} q^{4}}$ **c.** $\left(3cd^4\right)^3 = 3^3 c^3 \left(d^4\right)^3 = 27 c^{3} d^{12}$ **d.** $\left(\frac{a^4}{3n^7}\right)^2 = \frac{\left(a^4\right)^2}{3^2 \left(n^7\right)^2} = \frac{a^{8}}{9 n^{14}}$ **e.** $\left(\frac{2x^2}{y^6}\right)^3 = \frac{2^3 \left(x^2\right)^3}{\left(y^6\right)^3} = \frac{8 x^{6}}{y^{18}}$ **f.** $\left(2s^3 t^4\right)^5 = 2^5 \left(s^3\right)^5 \left(t^4\right)^5 = 32 s^{15} t^{20}$ **g.** $\left(\frac{4g^5}{h^2}\right)^2 = \frac{4^2 \left(g^5\right)^2}{\left(h^2\right)^2} = \frac{16 g^{10}}{h^{4}}$ **h.** $\left(\frac{3x^2 y^3}{4z}\right)^4 = \frac{3^4 \left(x^2\right)^4 \left(y^3\right)^4}{4^4 z^4} = \frac{81 x^{8} y^{12}}{256 z^{4}}$ **Final answers:** - a) $9a^{5}$ - b) $\frac{16}{p^{8} q^{4}}$ - c) $27 c^{3} d^{12}$ - d) $\frac{a^{8}}{9 n^{14}}$ - e) $\frac{8 x^{6}}{y^{18}}$ - f) $32 s^{15} t^{20}$ - g) $\frac{16 g^{10}}{h^{4}}$ - h) $\frac{81 x^{8} y^{12}}{256 z^{4}}$