4. \( \begin{array}{ll}\text { (q) }\left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} & \text { (r) } 3\left(2 a^{3}\right)^{2} \times 2\left(3 a^{2}\right)^{3} \\ \text { (a) }\left(\frac{3 a^{4}}{5 b^{6}}\right)^{2} & \text { (b) }\left(\frac{16 x^{5} y}{8 x y^{4}}\right)^{3} \\ \text { (d) }\left(\frac{6 x^{7}}{12 x^{9}}\right)^{-2} & \text { (c) }\left(\frac{2 a^{3} \cdot 3 a^{2}}{6\left(a^{3}\right)^{2}}\right)^{2} \\ \text { (g) } \frac{\left(x^{-2} y^{4}\right)^{2}}{x^{2} y^{-3}} & \text { (h) } \frac{2\left(a^{-2} b^{2}\right)^{-3} \times(a b)}{\left(2 b^{-6}\right)^{2}}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{2a^{3}\times 3a^{2}}{\left(6\left(a^{3}\right)^{2}\right)}\right)^{2}\) - step1: Remove the parentheses: \(\left(\frac{2a^{3}\times 3a^{2}}{6\left(a^{3}\right)^{2}}\right)^{2}\) - step2: Multiply the exponents: \(\left(\frac{2a^{3}\times 3a^{2}}{6a^{3\times 2}}\right)^{2}\) - step3: Multiply by \(a^{-n}:\) \(\left(\frac{2a^{3}\times 3a^{2}\times a^{-3\times 2}}{6}\right)^{2}\) - step4: Multiply the numbers: \(\left(\frac{2a^{3}\times 3a^{2}\times a^{-6}}{6}\right)^{2}\) - step5: Reduce the fraction: \(\left(\frac{1}{a}\right)^{2}\) - step6: Evaluate the power: \(a^{-2}\) - step7: Simplify: \(\frac{1}{a^{2}}\) Calculate or simplify the expression \( (2*(a^(-2)*b^2)^(-3)*(a*b))/(2*b^(-6))^2 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)\right)}{\left(2b^{-6}\right)^{2}}\) - step1: Evaluate: \(\frac{\left(2\left(a^{-2}b^{2}\right)^{-3}ab\right)}{\left(2b^{-6}\right)^{2}}\) - step2: Remove the parentheses: \(\frac{2\left(a^{-2}b^{2}\right)^{-3}ab}{\left(2b^{-6}\right)^{2}}\) - step3: Factor the expression: \(\frac{2\left(a^{-2}b^{2}\right)^{-3}ab}{2^{2}b^{-12}}\) - step4: Reduce the fraction: \(\frac{\left(a^{-2}b^{2}\right)^{-3}ab^{13}}{2}\) - step5: Evaluate the power: \(\frac{a^{6}b^{-6}ab^{13}}{2}\) - step6: Simplify: \(\frac{a^{7}b^{7}}{2}\) Calculate or simplify the expression \( (16*x^5*y/(8*x*y^4))^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{16x^{5}y}{8xy^{4}}\right)^{3}\) - step1: Divide the terms: \(\left(\frac{2x^{4}}{y^{3}}\right)^{3}\) - step2: Rewrite the expression: \(\frac{\left(2x^{4}\right)^{3}}{\left(y^{3}\right)^{3}}\) - step3: Evaluate the power: \(\frac{8x^{12}}{y^{9}}\) Calculate or simplify the expression \( 3*(2*a^3)^2*2*(3*a^2)^3 \). Simplify the expression by following steps: - step0: Solution: \(3\left(2a^{3}\right)^{2}\times 2\left(3a^{2}\right)^{3}\) - step1: Multiply the terms: \(6\left(2a^{3}\right)^{2}\left(3a^{2}\right)^{3}\) - step2: Multiply the terms: \(24a^{6}\left(3a^{2}\right)^{3}\) - step3: Rewrite the expression: \(24a^{6}\times 27a^{6}\) - step4: Multiply the numbers: \(648a^{6}\times a^{6}\) - step5: Multiply the terms: \(648a^{12}\) Calculate or simplify the expression \( (3*x^4)^2*(2*x^2)^4 \). Simplify the expression by following steps: - step0: Solution: \(\left(3x^{4}\right)^{2}\left(2x^{2}\right)^{4}\) - step1: Rewrite the expression: \(9x^{8}\times 16x^{8}\) - step2: Multiply the numbers: \(144x^{8}\times x^{8}\) - step3: Multiply the terms: \(144x^{16}\) Calculate or simplify the expression \( (6*x^7/(12*x^9))^-2 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{6x^{7}}{12x^{9}}\right)-2\) - step1: Divide the terms: \(\frac{1}{2x^{2}}-2\) - step2: Reduce fractions to a common denominator: \(\frac{1}{2x^{2}}-\frac{2\times 2x^{2}}{2x^{2}}\) - step3: Transform the expression: \(\frac{1-2\times 2x^{2}}{2x^{2}}\) - step4: Multiply the terms: \(\frac{1-4x^{2}}{2x^{2}}\) Calculate or simplify the expression \( ((x^(-2)*y^4)^2)/(x^2*y^(-3)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(\left(x^{-2}y^{4}\right)^{2}\right)}{\left(x^{2}y^{-3}\right)}\) - step1: Evaluate: \(\frac{\left(x^{-2}y^{4}\right)^{2}}{\left(x^{2}y^{-3}\right)}\) - step2: Remove the parentheses: \(\frac{\left(x^{-2}y^{4}\right)^{2}}{x^{2}y^{-3}}\) - step3: Rewrite the expression: \(\frac{\left(x^{-2}y^{4}\right)^{2}}{\frac{x^{2}}{y^{3}}}\) - step4: Multiply by the reciprocal: \(\left(x^{-2}y^{4}\right)^{2}\times \frac{y^{3}}{x^{2}}\) - step5: Multiply the terms: \(\frac{\left(x^{-2}y^{4}\right)^{2}y^{3}}{x^{2}}\) - step6: Multiply the terms: \(\frac{x^{-4}y^{11}}{x^{2}}\) - step7: Reduce the fraction: \(\frac{y^{11}}{x^{6}}\) Calculate or simplify the expression \( (3*a^4/(5*b^6))^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{3a^{4}}{5b^{6}}\right)^{2}\) - step1: Rewrite the expression: \(\frac{\left(3a^{4}\right)^{2}}{\left(5b^{6}\right)^{2}}\) - step2: Evaluate the power: \(\frac{9a^{8}}{25b^{12}}\) Let's simplify each expression step by step: ### (q) \( (3x^{4})^{2} \cdot (2x^{2})^{4} \) 1. Calculate \( (3x^{4})^{2} = 9x^{8} \). 2. Calculate \( (2x^{2})^{4} = 16x^{8} \). 3. Multiply the results: \( 9x^{8} \cdot 16x^{8} = 144x^{16} \). **Result:** \( 144x^{16} \) --- ### (r) \( 3(2a^{3})^{2} \times 2(3a^{2})^{3} \) 1. Calculate \( (2a^{3})^{2} = 4a^{6} \). 2. Calculate \( (3a^{2})^{3} = 27a^{6} \). 3. Multiply: \( 3 \cdot 4a^{6} \cdot 2 \cdot 27a^{6} = 648a^{12} \). **Result:** \( 648a^{12} \) --- ### (a) \( \left(\frac{3a^{4}}{5b^{6}}\right)^{2} \) 1. Square the numerator: \( (3a^{4})^{2} = 9a^{8} \). 2. Square the denominator: \( (5b^{6})^{2} = 25b^{12} \). 3. Combine: \( \frac{9a^{8}}{25b^{12}} \). **Result:** \( \frac{9a^{8}}{25b^{12}} \) --- ### (b) \( \left(\frac{16x^{5}y}{8xy^{4}}\right)^{3} \) 1. Simplify the fraction: \( \frac{16x^{5}y}{8xy^{4}} = 2x^{4}y^{-3} \). 2. Cube the result: \( (2x^{4}y^{-3})^{3} = 8x^{12}y^{-9} = \frac{8x^{12}}{y^{9}} \). **Result:** \( \frac{8x^{12}}{y^{9}} \) --- ### (d) \( \left(\frac{6x^{7}}{12x^{9}}\right)^{-2} \) 1. Simplify the fraction: \( \frac{6x^{7}}{12x^{9}} = \frac{1}{2x^{2}} \). 2. Apply the negative exponent: \( \left(\frac{1}{2x^{2}}\right)^{-2} = \frac{1}{(1/2)^{2}(x^{2})^{2}} = \frac{1-4x^{2}}{2x^{2}} \). **Result:** \( \frac{1-4x^{2}}{2x^{2}} \) --- ### (c) \( \left(\frac{2a^{3} \cdot 3a^{2}}{6(a^{3})^{2}}\right)^{2} \) 1. Simplify the fraction: \( \frac{6a^{5}}{6a^{6}} = \frac{1}{a} \). 2. Square the result: \( \left(\frac{1}{a}\right)^{2} = \frac{1}{a^{2}} \). **Result:** \( \frac{1}{a^{2}} \) --- ### (g) \( \frac{(x^{-2}y^{4})^{2}}{x^{2}y^{-3}} \) 1. Square the numerator: \( (x^{-2}y^{4})^{2} = x^{-4}y^{8} \). 2. Combine: \( \frac{x^{-4}y^{8}}{x^{2}y^{-3}} = x^{-4-2}y^{8-(-3)} = x^{-6}y^{11} \). 3. Rewrite: \( \frac{y^{11}}{x^{6}} \). **Result:** \( \frac{y^{11}}{x^{6}} \) --- ### (h) \( \frac{2(a^{-2}b^{2})^{-3} \times (ab)}{(2b^{-6})^{2}} \) 1. Simplify the numerator: \( (a^{-2}b^{2})^{-3} = a^{6}b^{-6} \). 2. Combine: \( 2a^{6}b^{-6} \cdot ab = 2a^{7}b^{-5} \). 3. Simplify the denominator: \( (2b^{-6})^{2} = 4b^{-12} \). 4. Combine: \( \frac{2a^{7}b^{-5}}{4b^{-12}} = \frac{a^{7}b^{7}}{2} \). **Result:** \( \frac{a^{7}b^{7}}{2} \) --- ### Summary of Results: - (q) \( 144x^{16} \) - (r) \( 648a^{12} \) - (a) \( \frac{9a^{8}}{25b^{12}} \) - (b) \( \frac{8x^{12}}{y^{9}} \) - (d) \( \frac{1-4x^{2}}{2x^{2}} \) - (c) \( \frac{1}{a^{2}} \) - (g) \( \frac{y^{11}}{x^{6}} \) - (h) \( \frac{a^{7}b^{7}}{2} \)