i) \( \frac{a^{2}}{c} \times \frac{c^{2}}{a} \) k) \( 3\left(10^{2}\right)^{-2} \) m) \( 2\left(x^{2}\right)^{3} \times 3\left(x^{4}\right)^{-2} \) o) \( \frac{2^{0} \times 2^{1} \times 2^{2}}{2^{-2} \times 2^{-1}} \)

Let's simplify each expression step by step. ────────────────────────────── (i) Expression: (a²/c) × (c²/a) Step 1: Multiply the numerators and the denominators:   = (a² × c²) / (c × a) Step 2: Combine the like bases by subtracting exponents:   = a^(2-1) × c^(2-1)   = a¹ × c¹   = a · c ────────────────────────────── (k) Expression: 3(10²)⁻² Step 1: Simplify the power inside the parentheses:   (10²)⁻² = 10^(2×(-2)) = 10^(-4) Step 2: Multiply by 3:   = 3 × 10^(-4)   = 3/10⁴   = 3/10000 ────────────────────────────── (m) Expression: 2(x²)³ × 3(x⁴)⁻² Step 1: Simplify each part:   (x²)³ = x^(2×3) = x⁶   (x⁴)⁻² = x^(4×(-2)) = x^(–8) Step 2: Substitute back:   = 2x⁶ × 3x^(–8)   = (2×3) × x^(6–8)   = 6x^(–2) Step 3: Write using positive exponents:   = 6/x² ────────────────────────────── (o) Expression: (2⁰ × 2¹ × 2²) / (2^(–2) × 2^(–1)) Step 1: Combine the exponents in the numerator and denominator by addition:   Numerator exponent: 0 + 1 + 2 = 3, so numerator = 2³.   Denominator exponent: (–2) + (–1) = –3, so denominator = 2^(–3). Step 2: Use the quotient rule (subtract exponents):   = 2^(3 – (–3)) = 2^(3+3) = 2⁶ Step 3: Evaluate 2⁶:   = 64 ────────────────────────────── Summary of Answers: (i) a · c (k) 3/10000 (m) 6/x² (o) 64