**Question 1: Simplify using laws of indices** 1.i Simplify $ (b^4)^3 b^{-2} $ Step 1: Use the power of a power rule: $$ (b^4)^3 = b^{4\times3} = b^{12} $$ Step 2: Multiply by $ b^{-2} $: $$ b^{12} \times b^{-2} = b^{12-2} = b^{10} $$ 1.ii Simplify $ (-3x)^2 + 3x^2 $ Step 1: Expand the square: $$ (-3x)^2 = (-3)^2 \times x^2 = 9x^2 $$ Step 2: Add: $$ 9x^2 + 3x^2 = 12x^2 $$ 1.iii Simplify $$ \left(\frac{x}{y}\right)^2 x^2 y $$ Step 1: Expand the square: $$ \left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2} $$ Step 2: Multiply by $ x^2 y $: $$ \frac{x^2}{y^2} \times x^2 y = \frac{x^{2+2} y}{y^2} = \frac{x^4 y}{y^2} $$ Step 3: Simplify $ y $ terms: $$ \frac{y}{y^2} = y^{-1} $$ Final answer: $$ x^4 y^{-1} = \frac{x^4}{y} $$ 1.iv Simplify $$ \frac{a^{-3} b^7}{a^5 b^3} $$ Step 1: Apply laws of indices for division: $$ a^{-3-5} b^{7-3} = a^{-8} b^4 $$ Step 2: Rewrite negative exponent: $$ \frac{b^4}{a^8} $$ 1.v Simplify $$ \frac{4^{20} - 4^{18}}{4^{18}} $$ Step 1: Factor numerator: $$ 4^{18} (4^2 - 1) $$ Step 2: Rewrite expression: $$ \frac{4^{18} (16-1)}{4^{18}} = 16 - 1 = 15 $$ **Question 2: Fill in the blanks with set symbols** 2.i $ a \quad \in \quad \{a, b, c, d\} $ 2.ii $ \{4, 5\} \quad \subseteq \quad \{1, 2, 3, 4, 5\} $ 2.iii $ \{b, c\} \quad \in \quad \{a, \{b, c\}, d\} $ 2.iv $ -4 \quad \notin \quad \{1, 2, 3, 4, 5\} $ 2.v $ \{1, -\frac{2}{3}\} \quad = \quad \{ x \mid 3x^2 - x - 2=0 \text{ and } x \text{ is a real number} \} $ Final answers: 1.i $b^{10}$ 1.ii $12x^2$ 1.iii $\frac{x^4}{y}$ 1.iv $\frac{b^4}{a^8}$ 1.v $15$ 2.i $\in$ 2.ii $\subseteq$ 2.iii $\in$ 2.iv $\notin$ 2.v $=$