12. Simplify \( \begin{array}{ll}\text { a) }\left(x^{4} y^{3}\right)^{2}-x y^{3} \times x^{7} y^{3} & \text { ()) } \frac{x^{3} y^{2}}{2^{-3}} \\ \text { c) } 24 x^{2} b^{2}+a^{2} \times 3 a^{4} b^{2} \\ 12 x y^{3}-x y & \text { d) } 12 b^{4} c^{2}+3 b^{3}-4 b c^{2}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(12b^{4}c^{2}+3b^{3}-4bc^{2}\) Calculate or simplify the expression \( (x^4 * y^3)^2 - (x * y^3) * (x^7 * y^3) \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{4}y^{3}\right)^{2}-xy^{3}x^{7}y^{3}\) - step1: Multiply the terms: \(\left(x^{4}y^{3}\right)^{2}-x^{8}y^{6}\) - step2: Rewrite the expression: \(x^{8}y^{6}-x^{8}y^{6}\) - step3: Subtract the terms: \(0\) Calculate or simplify the expression \( (x^3 * y^2) / (2^(-3)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x^{3}y^{2}\right)}{\left(2^{-3}\right)}\) - step1: Evaluate: \(\frac{x^{3}y^{2}}{2^{-3}}\) - step2: Rewrite the expression: \(\frac{x^{3}y^{2}}{\frac{1}{2^{3}}}\) - step3: Multiply by the reciprocal: \(x^{3}y^{2}\times 2^{3}\) - step4: Reorder the terms: \(2^{3}x^{3}y^{2}\) - step5: Evaluate the power: \(8x^{3}y^{2}\) Calculate or simplify the expression \( 24 * x^2 * b^2 + (a^2 * 3 * a^4 * b^2) \). Simplify the expression by following steps: - step0: Solution: \(24x^{2}b^{2}+\left(a^{2}\times 3a^{4}b^{2}\right)\) - step1: Multiply: \(24x^{2}b^{2}+3a^{6}b^{2}\) Calculate or simplify the expression \( 12 * x * y^3 - (x * y) \). Simplify the expression by following steps: - step0: Solution: \(12xy^{3}-xy\) Let's simplify each expression step by step: ### a) \( \left(x^{4} y^{3}\right)^{2}-x y^{3} \times x^{7} y^{3} \) 1. **Calculate \( \left(x^{4} y^{3}\right)^{2} \)**: \[ (x^4 y^3)^2 = x^{4 \cdot 2} y^{3 \cdot 2} = x^8 y^6 \] 2. **Calculate \( x y^{3} \times x^{7} y^{3} \)**: \[ x y^3 \times x^7 y^3 = x^{1 + 7} y^{3 + 3} = x^8 y^6 \] 3. **Combine the results**: \[ x^8 y^6 - x^8 y^6 = 0 \] **Result**: \( 0 \) --- ### b) \( \frac{x^{3} y^{2}}{2^{-3}} \) 1. **Calculate \( 2^{-3} \)**: \[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \] 2. **Rewrite the expression**: \[ \frac{x^3 y^2}{2^{-3}} = x^3 y^2 \times 2^3 = x^3 y^2 \times 8 = 8 x^3 y^2 \] **Result**: \( 8 x^{3} y^{2} \) --- ### c) \( 24 x^{2} b^{2}+a^{2} \times 3 a^{4} b^{2} \) 1. **Calculate \( a^{2} \times 3 a^{4} b^{2} \)**: \[ a^2 \times 3 a^4 b^2 = 3 a^{2 + 4} b^2 = 3 a^6 b^2 \] 2. **Combine the results**: \[ 24 x^2 b^2 + 3 a^6 b^2 \] **Result**: \( 24 x^{2} b^{2} + 3 a^{6} b^{2} \) --- ### d) \( 12 x y^{3}-x y \) 1. **Factor out \( x \)**: \[ 12 x y^3 - x y = x(12 y^3 - y) \] 2. **Factor out \( y \) from the expression in parentheses**: \[ 12 y^3 - y = y(12 y^2 - 1) \] 3. **Final expression**: \[ x y (12 y^2 - 1) \] **Result**: \( 12 x y^{3} - x y \) --- ### e) \( 12 b^{4} c^{2}+3 b^{3}-4 b c^{2} \) 1. **Combine like terms**: \[ 12 b^4 c^2 + 3 b^3 - 4 b c^2 \] **Result**: \( 12 b^{4} c^{2} + 3 b^{3} - 4 b c^{2} \) --- ### Summary of Results: - a) \( 0 \) - b) \( 8 x^{3} y^{2} \) - c) \( 24 x^{2} b^{2} + 3 a^{6} b^{2} \) - d) \( 12 x y^{3} - x y \) - e) \( 12 b^{4} c^{2} + 3 b^{3} - 4 b c^{2} \)