1. Simplify. \( \begin{array}{llll}\text { a) }(\sqrt{26})^{2} & \text { b) }(\sqrt{1200})^{2} & \text { c) }(\sqrt[3]{216})^{3} & \text { d) }(\sqrt[3]{1000})^{3} \\ \text { e) }(\sqrt[3]{125000})^{3} & \text { f) }(\sqrt[3]{32})^{5} & \text { g) }(\sqrt[6]{64})^{6} & \text { h) }(\sqrt[10]{1024})^{10}\end{array} \) 2. Simplify. \( \begin{array}{ll}\text { a) } \sqrt{36} \times 2^{3}-3^{2} & \text { b) } 3^{2}-\sqrt[3]{27} \\ \text { c) } 2 \sqrt{4}+3^{2}-\left(2^{3}\right)^{2} & \text { d) } 9^{2}+\sqrt{10^{2}}-13 \\ \text { e) } \sqrt{120+24}-\sqrt{130-9} & \text { f) } 6^{4} \times 6^{5} \times 6^{1} \div 6^{7} \div 6^{3}\end{array} \)

Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[6]{64}\right)^{6}\) - step1: Simplify the root: \(2^{6}\) - step2: Evaluate the power: \(64\) Calculate or simplify the expression \( 3^2 - \sqrt[3](27) \). Calculate the value by following steps: - step0: Calculate: \(3^{2}-\sqrt[3]{27}\) - step1: Simplify the root: \(3^{2}-3\) - step2: Evaluate the power: \(9-3\) - step3: Subtract the numbers: \(6\) Calculate or simplify the expression \( 6^4 * 6^5 * 6^1 / 6^7 / 6^3 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\frac{6^{4}\times 6^{5}\times 6^{1}}{6^{7}}}{6^{3}}\) - step1: Multiply by \(a^{-n}:\) \(\frac{6^{4}\times 6^{5}\times 6\times 6^{-7}}{6^{3}}\) - step2: Multiply by \(a^{-n}:\) \(6^{4}\times 6^{5}\times 6\times 6^{-7}\times 6^{-3}\) - step3: Multiply the terms: \(6^{4+5+1-7-3}\) - step4: Calculate: \(6^{0}\) - step5: Evaluate the power: \(1\) Calculate or simplify the expression \( (\sqrt[10](1024))^10 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[10]{1024}\right)^{10}\) - step1: Simplify the root: \(2^{10}\) - step2: Evaluate the power: \(1024\) Calculate or simplify the expression \( 9^2 + \sqrt(10^2) - 13 \). Calculate the value by following steps: - step0: Calculate: \(9^{2}+\sqrt{10^{2}}-13\) - step1: Simplify the root: \(9^{2}+10-13\) - step2: Evaluate the power: \(81+10-13\) - step3: Calculate: \(78\) Calculate or simplify the expression \( 2 * \sqrt(4) + 3^2 - (2^3)^2 \). Calculate the value by following steps: - step0: Calculate: \(2\sqrt{4}+3^{2}-\left(2^{3}\right)^{2}\) - step1: Multiply the exponents: \(2\sqrt{4}+3^{2}-2^{3\times 2}\) - step2: Simplify the root: \(2\times 2+3^{2}-2^{3\times 2}\) - step3: Multiply the numbers: \(2\times 2+3^{2}-2^{6}\) - step4: Multiply the numbers: \(4+3^{2}-2^{6}\) - step5: Evaluate the power: \(4+9-2^{6}\) - step6: Evaluate the power: \(4+9-64\) - step7: Calculate: \(-51\) Calculate or simplify the expression \( \sqrt(36) * 2^3 - 3^2 \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{36}\times 2^{3}-3^{2}\) - step1: Simplify the root: \(6\times 2^{3}-3^{2}\) - step2: Multiply the terms: \(48-3^{2}\) - step3: Evaluate the power: \(48-9\) - step4: Subtract the numbers: \(39\) Calculate or simplify the expression \( \sqrt(120 + 24) - \sqrt(130 - 9) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{120+24}-\sqrt{130-9}\) - step1: Add the numbers: \(\sqrt{144}-\sqrt{130-9}\) - step2: Simplify the root: \(12-\sqrt{130-9}\) - step3: Subtract the numbers: \(12-\sqrt{121}\) - step4: Simplify the root: \(12-11\) - step5: Subtract the numbers: \(1\) Calculate or simplify the expression \( (\sqrt[3](216))^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[3]{216}\right)^{3}\) - step1: Simplify the root: \(6^{3}\) - step2: Evaluate the power: \(216\) Calculate or simplify the expression \( (\sqrt(26))^2 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt{26}\right)^{2}\) - step1: Simplify the root: \(26\) Calculate or simplify the expression \( (\sqrt[3](1000))^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[3]{1000}\right)^{3}\) - step1: Simplify the root: \(10^{3}\) - step2: Evaluate the power: \(1000\) Calculate or simplify the expression \( (\sqrt[3](125000))^3 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[3]{125000}\right)^{3}\) - step1: Simplify the root: \(50^{3}\) Calculate or simplify the expression \( (\sqrt(1200))^2 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt{1200}\right)^{2}\) - step1: Simplify the root: \(\left(20\sqrt{3}\right)^{2}\) - step2: Evaluate the power: \(400\times 3\) - step3: Multiply the numbers: \(1200\) Calculate or simplify the expression \( (\sqrt[3](32))^5 \). Calculate the value by following steps: - step0: Calculate: \(\left(\sqrt[3]{32}\right)^{5}\) - step1: Simplify the root: \(\left(2\sqrt[3]{4}\right)^{5}\) - step2: Evaluate the power: \(32\times 2^{3}\sqrt[3]{2}\) - step3: Multiply the terms: \(2^{8}\sqrt[3]{2}\) - step4: Evaluate the power: \(256\sqrt[3]{2}\) Let's simplify the expressions step by step. ### Part 1: Simplifying the Expressions 1. **a)** \( (\sqrt{26})^{2} \) - Simplification: \( 26 \) 2. **b)** \( (\sqrt{1200})^{2} \) - Simplification: \( 1200 \) 3. **c)** \( (\sqrt[3]{216})^{3} \) - Simplification: \( 216 \) 4. **d)** \( (\sqrt[3]{1000})^{3} \) - Simplification: \( 1000 \) 5. **e)** \( (\sqrt[3]{125000})^{3} \) - Simplification: \( 125000 \) or \( 50^{3} \) (both are valid, but \( 125000 \) is more straightforward) 6. **f)** \( (\sqrt[3]{32})^{5} \) - Simplification: \( 256\sqrt[3]{2} \) 7. **g)** \( (\sqrt[6]{64})^{6} \) - Simplification: \( 64 \) 8. **h)** \( (\sqrt[10]{1024})^{10} \) - Simplification: \( 1024 \) ### Part 2: Further Simplifying the Expressions 1. **a)** \( \sqrt{36} \times 2^{3} - 3^{2} \) - Calculation: \( 6 \times 8 - 9 = 48 - 9 = 39 \) 2. **b)** \( 3^{2} - \sqrt[3]{27} \) - Calculation: \( 9 - 3 = 6 \) 3. **c)** \( 2 \sqrt{4} + 3^{2} - (2^{3})^{2} \) - Calculation: \( 2 \times 2 + 9 - 16 = 4 + 9 - 16 = -3 \) 4. **d)** \( 9^{2} + \sqrt{10^{2}} - 13 \) - Calculation: \( 81 + 10 - 13 = 81 + 10 - 13 = 78 \) 5. **e)** \( \sqrt{120 + 24} - \sqrt{130 - 9} \) - Calculation: \( \sqrt{144} - \sqrt{121} = 12 - 11 = 1 \) 6. **f)** \( 6^{4} \times 6^{5} \times 6^{1} \div 6^{7} \div 6^{3} \) - Calculation: \( 6^{4 + 5 + 1 - 7 - 3} = 6^{0} = 1 \) ### Summary of Results **Part 1 Results:** - a) 26 - b) 1200 - c) 216 - d) 1000 - e) 125000 - f) \( 256\sqrt[3]{2} \) - g) 64 - h) 1024 **Part 2 Results:** - a) 39 - b) 6 - c) -3 - d) 78 - e) 1 - f) 1