(a) \( \left(x^{3}\right)^{4} \) (b) \( \quad\left(x^{2}\right)^{5} \) (c) \( \quad\left(x^{2}\right)^{2} \) (d) \( \quad(x)^{3} \) (e) \( \left(3^{2}\right)^{3} \) (f) \( \quad\left(5^{4}\right)^{4} \) (g) \( \quad\left(4 x^{3}\right)^{3} \) (h) \( \quad\left(7 a^{7}\right)^{2} \) (i) \( \left(-4 x^{3}\right)^{3} \) (j) \( \quad\left(-4 x^{3}\right)^{4} \) (k) \( \left(x^{4}\right)^{2}+\left(x^{2}\right)^{4} \) (l) \( \quad 3\left(x^{4}\right)^{2}+2\left(x^{2}\right)^{4} \) (m) \( \quad\left(x^{4}\right)^{2} \cdot\left(x^{2}\right)^{4} \) (n) \( \quad 3\left(x^{4}\right)^{2} \times 2\left(x^{2}\right)^{4} \) (o) \( \quad\left(3 x^{4}\right)^{2}+\left(2 x^{2}\right)^{4} \) (p) \( \quad 3\left(2 a^{3}\right)^{2}+2\left(3 a^{2}\right)^{3} \) (q) \( \left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} \) (r) \( 3\left(2 a^{3}\right)^{2} \times 2\left(3 a^{2}\right)^{3} \) Simplify the following: (a) \( \left(\frac{3 a^{4}}{5 b^{6}}\right)^{2} \) (b) \( \left(\frac{16 x^{5} y}{8 x y^{4}}\right)^{3} \) (c) \( \quad\left(\frac{3 p^{3} q^{2}}{9 p^{5}}\right. \) (d) \( \left(\frac{6 x^{7}}{12 x^{9}}\right)^{-2} \) (e) \( \left(\frac{2 a^{3} \cdot 3 a^{2}}{6\left(a^{3}\right)^{2}}\right)^{2} \) (f) \( \quad\left(\frac{x^{3}+x^{3}}{x^{3} \times x^{3}}\right. \) (g) \( \frac{\left(x^{-2} y^{4}\right)^{2}}{x^{2} y^{-3}} \) (h) \( \frac{2\left(a^{-2} b^{2}\right)^{-3} \times(a b)^{-6}}{\left(2 b^{-6}\right)^{2}} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{6x^{7}}{\left(12x^{9}\right)}\right)^{-2}\) - step1: Evaluate: \(\left(\frac{6x^{7}}{12x^{9}}\right)^{-2}\) - step2: Divide the terms: \(\left(\frac{1}{2x^{2}}\right)^{-2}\) - step3: Evaluate the power: \(\left(2x^{2}\right)^{2}\) - step4: Use the properties of exponents: \(2^{2}\left(x^{2}\right)^{2}\) - step5: Evaluate the power: \(4x^{4}\) Calculate or simplify the expression \( ((x^3+x^3)/(x^3*x^3)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x^{3}+x^{3}\right)}{\left(x^{3}\times x^{3}\right)}\) - step1: Remove the parentheses: \(\frac{x^{3}+x^{3}}{x^{3}\times x^{3}}\) - step2: Multiply the terms: \(\frac{x^{3}+x^{3}}{x^{6}}\) - step3: Add the terms: \(\frac{2x^{3}}{x^{6}}\) - step4: Reduce the fraction: \(\frac{2}{x^{6-3}}\) - step5: Reduce the fraction: \(\frac{2}{x^{3}}\) Calculate or simplify the expression \( (2*(a^(-2)*b^2)^(-3)*(a*b)^(-6))/((2*b^(-6))^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)^{-6}\right)}{\left(\left(2b^{-6}\right)^{2}\right)}\) - step1: Evaluate: \(\frac{\left(2\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)^{-6}\right)}{\left(2b^{-6}\right)^{2}}\) - step2: Remove the parentheses: \(\frac{2\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)^{-6}}{\left(2b^{-6}\right)^{2}}\) - step3: Factor the expression: \(\frac{2\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)^{-6}}{2^{2}b^{-12}}\) - step4: Reduce the fraction: \(\frac{\left(a^{-2}b^{2}\right)^{-3}\left(ab\right)^{-6}}{2b^{-12}}\) - step5: Evaluate the power: \(\frac{a^{6}b^{-6}\left(ab\right)^{-6}}{2b^{-12}}\) - step6: Use the properties of exponents: \(\frac{a^{6}b^{-6}a^{-6}b^{-6}}{2b^{-12}}\) - step7: Simplify: \(\frac{b^{-12}}{2b^{-12}}\) - step8: Reduce the fraction: \(\frac{1}{2}\) 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 \( (2*a^3*3*a^2/(6*(a^3)^2))^2 \). 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 \( (16*x^5*y/(8*x*y^4))^3 \). 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 \( (3*a^4/5*b^6)^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{3a^{4}}{5}b^{6}\right)^{2}\) - step1: Multiply the terms: \(\left(\frac{3a^{4}b^{6}}{5}\right)^{2}\) - step2: Rewrite the expression: \(\frac{\left(3a^{4}b^{6}\right)^{2}}{5^{2}}\) - step3: Evaluate the power: \(\frac{9a^{8}b^{12}}{25}\) Calculate or simplify the expression \( (3*p^3*q^2/(9*p^5)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{3p^{3}q^{2}}{9p^{5}}\) - step1: Reduce the fraction: \(\frac{3q^{2}}{9p^{5-3}}\) - step2: Reduce the fraction: \(\frac{3q^{2}}{9p^{2}}\) - step3: Reduce the fraction: \(\frac{q^{2}}{3p^{2}}\) Here are the simplified results for each of the expressions provided: ### Simplifications: (a) \[ \left(\frac{3 a^{4}}{5 b^{6}}\right)^{2} = \frac{9a^{8}}{25b^{12}} \] (b) \[ \left(\frac{16 x^{5} y}{8 x y^{4}}\right)^{3} = \frac{8x^{12}}{y^{9}} \] (c) \[ \left(\frac{3 p^{3} q^{2}}{9 p^{5}}\right) = \frac{q^{2}}{3p^{2}} \] (d) \[ \left(\frac{6 x^{7}}{12 x^{9}}\right)^{-2} = 4x^{4} \] (e) \[ \left(\frac{2 a^{3} \cdot 3 a^{2}}{6\left(a^{3}\right)^{2}}\right)^{2} = \frac{1}{a^{2}} \] (f) \[ \left(\frac{x^{3}+x^{3}}{x^{3} \times x^{3}}\right) = \frac{2}{x^{3}} \] (g) \[ \frac{\left(x^{-2} y^{4}\right)^{2}}{x^{2} y^{-3}} = \frac{y^{11}}{x^{6}} \] (h) \[ \frac{2\left(a^{-2} b^{2}\right)^{-3} \times(a b)^{-6}}{\left(2 b^{-6}\right)^{2}} = \frac{1}{2} \] These results provide the simplified forms of the given expressions. 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