(c) Simplify the following, leaving your answers with positive exponenis. (1)* \( \left(x^{-3}\right)^{4} \) (2) \( \quad\left(x^{2}\right)^{-5} \) (3)* \( \left(x^{-2}\right)^{-2} \) (4) \( \left(-4 x^{-3}\right)^{3} \) (5) \( \quad\left(-2 x^{3}\right)^{4} \) (6) \( \left(x^{4}\right)^{2}+\left(x^{2}\right)^{4} \) (7) * \( 2\left(x^{4}\right)^{2}+4\left(x^{2}\right)^{4} \) (8) \( \quad\left(-x^{3}\right)^{2} \cdot\left(-x^{2}\right)^{3} \) \( (9) \star \quad 2\left(-x^{4}\right)^{2} \times 4\left(-x^{2}\right)^{4} \) (10) \( \left(3 x^{4}\right)^{2}-\left(2 x^{2}\right)^{4} \) (11)* \( 3\left(-2 a^{3}\right)^{2}-2\left(-3 a^{2}\right)^{3} \) (12) \( \left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} \) \( (13) * 3\left(-2 a^{3}\right)^{2} \times 2\left(-3 a^{2}\right)^{3} \) (d) Simplify the following, leaving your answers with positive exponents: \( (1)^{*}\left(\frac{a^{4}}{b^{6}}\right)^{2} \) (2) \( \left(\frac{10 x^{3} y}{5 x y^{2}}\right)^{3} \) (3) \( \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}}\right)^{3} \) (4) \( \left(\frac{3 x^{-5}}{9 x^{7}}\right)^{-2} \) (5)* \( \left(\frac{2 a^{3} \cdot 4 a^{2}}{8\left(a^{-3}\right)^{2}}\right)^{2} \) (6) \( )\left(\frac{x^{2}+x^{2}}{x^{2} \times x^{2}}\right)^{2} \)

Simplify the expression by following steps: - step0: Solution: \(2\left(x^{4}\right)^{2}\times 4\left(x^{2}\right)^{4}\) - step1: Multiply the exponents: \(2\left(x^{4}\right)^{2}\times 4x^{2\times 4}\) - step2: Multiply the exponents: \(2x^{4\times 2}\times 4x^{2\times 4}\) - step3: Multiply the numbers: \(2x^{8}\times 4x^{2\times 4}\) - step4: Multiply the numbers: \(2x^{8}\times 4x^{8}\) - step5: Multiply the terms: \(8x^{8}\times x^{8}\) - step6: Multiply the terms: \(8x^{8+8}\) - step7: Add the numbers: \(8x^{16}\) Calculate or simplify the expression \( ((x^{2}+x^{2})/(x^{2} * x^{2}))^{2} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{\left(x^{2}+x^{2}\right)}{\left(x^{2}\times x^{2}\right)}\right)^{2}\) - step1: Remove the parentheses: \(\left(\frac{x^{2}+x^{2}}{x^{2}\times x^{2}}\right)^{2}\) - step2: Multiply the terms: \(\left(\frac{x^{2}+x^{2}}{x^{4}}\right)^{2}\) - step3: Add the terms: \(\left(\frac{2x^{2}}{x^{4}}\right)^{2}\) - step4: Divide the terms: \(\left(\frac{2}{x^{2}}\right)^{2}\) - step5: Rewrite the expression: \(\frac{2^{2}}{\left(x^{2}\right)^{2}}\) - step6: Evaluate the power: \(\frac{4}{x^{4}}\) Calculate or simplify the expression \( (a^{4}/b^{6})^{2} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{a^{4}}{b^{6}}\right)^{2}\) - step1: Rewrite the expression: \(\frac{\left(a^{4}\right)^{2}}{\left(b^{6}\right)^{2}}\) - step2: Evaluate the power: \(\frac{a^{8}}{b^{12}}\) Calculate or simplify the expression \( (-4 x^{-3})^{3} \). Simplify the expression by following steps: - step0: Solution: \(\left(-4x^{-3}\right)^{3}\) - step1: Determine the sign: \(-\left(4x^{-3}\right)^{3}\) - step2: Use the properties of exponents: \(-4^{3}\left(x^{-3}\right)^{3}\) - step3: Evaluate the power: \(-64x^{-9}\) - step4: Simplify: \(-\frac{64}{x^{9}}\) Calculate or simplify the expression \( (x^{-3})^{4} \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{-3}\right)^{4}\) - step1: Multiply the exponents: \(x^{-3\times 4}\) - step2: Multiply the numbers: \(x^{-12}\) - step3: Simplify: \(\frac{1}{x^{12}}\) Calculate or simplify the expression \( (x^{2})^{-5} \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{2}\right)^{-5}\) - step1: Multiply the exponents: \(x^{2\left(-5\right)}\) - step2: Multiply the numbers: \(x^{-10}\) - step3: Simplify: \(\frac{1}{x^{10}}\) Calculate or simplify the expression \( (x^{-2})^{-2} \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{-2}\right)^{-2}\) - step1: Multiply the exponents: \(x^{-2\left(-2\right)}\) - step2: Multiply the numbers: \(x^{4}\) Calculate or simplify the expression \( (x^{4})^{2}+(x^{2})^{4} \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{4}\right)^{2}+\left(x^{2}\right)^{4}\) - step1: Multiply the exponents: \(\left(x^{4}\right)^{2}+x^{2\times 4}\) - step2: Multiply the exponents: \(x^{4\times 2}+x^{2\times 4}\) - step3: Multiply the numbers: \(x^{8}+x^{2\times 4}\) - step4: Multiply the numbers: \(x^{8}+x^{8}\) - step5: Collect like terms: \(\left(1+1\right)x^{8}\) - step6: Add the numbers: \(2x^{8}\) Calculate or simplify the expression \( (-2 x^{3})^{4} \). Simplify the expression by following steps: - step0: Solution: \(\left(-2x^{3}\right)^{4}\) - step1: Determine the sign: \(\left(2x^{3}\right)^{4}\) - step2: Use the properties of exponents: \(2^{4}\left(x^{3}\right)^{4}\) - step3: Evaluate the power: \(16x^{12}\) Calculate or simplify the expression \( 2(x^{4})^{2}+4(x^{2})^{4} \). Simplify the expression by following steps: - step0: Solution: \(2\left(x^{4}\right)^{2}+4\left(x^{2}\right)^{4}\) - step1: Multiply the exponents: \(2\left(x^{4}\right)^{2}+4x^{2\times 4}\) - step2: Multiply the exponents: \(2x^{4\times 2}+4x^{2\times 4}\) - step3: Multiply the numbers: \(2x^{8}+4x^{2\times 4}\) - step4: Multiply the numbers: \(2x^{8}+4x^{8}\) - step5: Collect like terms: \(\left(2+4\right)x^{8}\) - step6: Add the numbers: \(6x^{8}\) Calculate or simplify the expression \( (10 x^{3} y / (5 x y^{2}))^{3} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{10x^{3}y}{5xy^{2}}\right)^{3}\) - step1: Divide the terms: \(\left(\frac{2x^{2}}{y}\right)^{3}\) - step2: Rewrite the expression: \(\frac{\left(2x^{2}\right)^{3}}{y^{3}}\) - step3: Evaluate the power: \(\frac{8x^{6}}{y^{3}}\) Calculate or simplify the expression \( ((-2 p^{4} q^{-2})/(4 p^{-6}))^{3} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{\left(-2p^{4}q^{-2}\right)}{\left(4p^{-6}\right)}\right)^{3}\) - step1: Remove the parentheses: \(\left(\frac{-2p^{4}q^{-2}}{4p^{-6}}\right)^{3}\) - step2: Multiply by \(a^{-n}:\) \(\left(\frac{-2p^{4}q^{-2}p^{6}}{4}\right)^{3}\) - step3: Reduce the fraction: \(\left(-\frac{p^{10}q^{-2}}{2}\right)^{3}\) - step4: Rewrite the expression: \(\left(-\frac{p^{10}}{2q^{2}}\right)^{3}\) - step5: Rewrite the expression: \(\frac{\left(-p^{10}\right)^{3}}{\left(2q^{2}\right)^{3}}\) - step6: Evaluate the power: \(\frac{-p^{30}}{8q^{6}}\) - step7: Rewrite the fraction: \(-\frac{p^{30}}{8q^{6}}\) Calculate or simplify the expression \( 3(-2 a^{3})^{2}-2(-3 a^{2})^{3} \). Simplify the expression by following steps: - step0: Solution: \(3\left(-2a^{3}\right)^{2}-2\left(-3a^{2}\right)^{3}\) - step1: Multiply the terms: \(12a^{6}-2\left(-3a^{2}\right)^{3}\) - step2: Multiply the terms: \(12a^{6}+54a^{6}\) - step3: Collect like terms: \(\left(12+54\right)a^{6}\) - step4: Add the numbers: \(66a^{6}\) 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: \(9x^{8}-\left(2x^{2}\right)^{4}\) - step2: Rewrite the expression: \(9x^{8}-16x^{8}\) - step3: Collect like terms: \(\left(9-16\right)x^{8}\) - step4: Subtract the numbers: \(-7x^{8}\) 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 \( (-x^{3})^{2} * (-x^{2})^{3} \). Simplify the expression by following steps: - step0: Solution: \(\left(x^{3}\right)^{2}\left(x^{2}\right)^{3}\) - step1: Multiply the exponents: \(\left(x^{3}\right)^{2}x^{2\times 3}\) - step2: Multiply the exponents: \(x^{3\times 2}\times x^{2\times 3}\) - step3: Multiply the numbers: \(x^{6}\times x^{2\times 3}\) - step4: Multiply the numbers: \(x^{6}\times x^{6}\) - step5: Simplify the expression: \(x^{6+6}\) - step6: Add the numbers: \(x^{12}\) Calculate or simplify the expression \( (3 x^{-5}/(9 x^{7}))^{-2} \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{3x^{-5}}{\left(9x^{7}\right)}\right)^{-2}\) - step1: Evaluate: \(\left(\frac{3x^{-5}}{9x^{7}}\right)^{-2}\) - step2: Reduce the fraction: \(\left(\frac{1}{3x^{12}}\right)^{-2}\) - step3: Evaluate the power: \(\left(3x^{12}\right)^{2}\) - step4: Use the properties of exponents: \(3^{2}\left(x^{12}\right)^{2}\) - step5: Evaluate the power: \(9x^{24}\) Calculate or simplify the expression \( (2 a^{3} * 4 a^{2})/(8(a^{-3})^{2}) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2a^{3}\times 4a^{2}\right)}{\left(8\left(a^{-3}\right)^{2}\right)}\) - step1: Remove the parentheses: \(\frac{2a^{3}\times 4a^{2}}{8\left(a^{-3}\right)^{2}}\) - step2: Multiply the exponents: \(\frac{2a^{3}\times 4a^{2}}{8a^{-3\times 2}}\) - step3: Multiply by \(a^{-n}:\) \(\frac{2a^{3}\times 4a^{2}\times a^{3\times 2}}{8}\) - step4: Multiply the numbers: \(\frac{2a^{3}\times 4a^{2}\times a^{6}}{8}\) - step5: Multiply: \(\frac{8a^{11}}{8}\) - step6: Reduce the fraction: \(a^{11}\) 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: \(24a^{6}\left(-27a^{6}\right)\) - step4: Multiply the numbers: \(-648a^{6}\times a^{6}\) - step5: Multiply the terms: \(-648a^{12}\) Here are the simplified results for each expression, leaving answers with positive exponents: ### Part (c) 1. \( \left(x^{-3}\right)^{4} = \frac{1}{x^{12}} \) 2. \( \left(x^{2}\right)^{-5} = \frac{1}{x^{10}} \) 3. \( \left(x^{-2}\right)^{-2} = x^{4} \) 4. \( \left(-4 x^{-3}\right)^{3} = -\frac{64}{x^{9}} \) 5. \( \left(-2 x^{3}\right)^{4} = 16x^{12} \) 6. \( \left(x^{4}\right)^{2}+\left(x^{2}\right)^{4} = 2x^{8} \) 7. \( 2\left(x^{4}\right)^{2}+4\left(x^{2}\right)^{4} = 6x^{8} \) 8. \( \left(-x^{3}\right)^{2} \cdot\left(-x^{2}\right)^{3} = x^{12} \) 9. \( 2\left(-x^{4}\right)^{2} \times 4\left(-x^{2}\right)^{4} = 8x^{16} \) 10. \( \left(3 x^{4}\right)^{2}-\left(2 x^{2}\right)^{4} = -7x^{8} \) 11. \( 3\left(-2 a^{3}\right)^{2}-2\left(-3 a^{2}\right)^{3} = 66a^{6} \) 12. \( \left(3 x^{4}\right)^{2} \cdot\left(2 x^{2}\right)^{4} = 144x^{16} \) 13. \( 3\left(-2 a^{3}\right)^{2} \times 2\left(-3 a^{2}\right)^{3} = -648a^{12} \) ### Part (d) 1. \( \left(\frac{a^{4}}{b^{6}}\right)^{2} = \frac{a^{8}}{b^{12}} \) 2. \( \left(\frac{10 x^{3} y}{5 x y^{2}}\right)^{3} = \frac{8x^{6}}{y^{3}} \) 3. \( \left(\frac{-2 p^{4} q^{-2}}{4 p^{-6}}\right)^{3} = -\frac{p^{30}}{8q^{6}} \) 4. \( \left(\frac{3 x^{-5}}{9 x^{7}}\right)^{-2} = 9x^{24} \) 5. \( \left(\frac{2 a^{3} \cdot 4 a^{2}}{8\left(a^{-3}\right)^{2}}\right)^{2} = a^{11} \) 6. \( \left(\frac{x^{2}+x^{2}}{x^{2} \times x^{2}}\right)^{2} = \frac{4}{x^{4}} \) If you have any further questions or need additional assistance, feel free to ask!