\( \left. \begin{array} { l } { 2.1 ( - 6 x ^ { 2 } ) ^ { 3 } \times ( - x ^ { - 3 } ) ^ { - 2 } \div ( - 3 x ^ { 3 } ) ^ { 2 } } \\ { 2.23 .5 ^ { x } + 6.5 ^ { x } - \frac { 4 } { 5 ^ { - x } } } \\ { 2.3 \frac { 7 ^ { n + 1 } \cdot 14 ^ { n - 1 } } { 98 ^ { n - 1 } } } \\ { 2.4 \frac { 25 ^ { n - 1 } \cdot 6 ^ { n } } { 10 ^ { n - 1 } \cdot 15 ^ { n } } } \\ { 2.5 \frac { 2 ^ { x - 1 } + 2 ^ { x + 3 } } { 3.2 ^ { x + 1 } } } \\ { 2.6 \frac { 5 ^ { x + 2 } - 4.5 ^ { x } } { 5 ^ { x - 1 } + 2.5 ^ { x + 1 } } } \\ { 2.7 \frac { 5 ^ { 1000 } - 5 ^ { 502 } + 24 } { 5 ^ { 501 } - 5 } } \\ { \sqrt { 1 a + 1 } \times \sqrt[ 3 ] { 8 ^ { a + 1 } } } \end{array} \right. \)

Simplify the expression by following steps: - step0: Solution: \(\sqrt{1\times a+1}\times \left(8^{a+1}\right)^{\frac{1}{3}}\) - step1: Multiply: \(\sqrt{a+1}\times \left(8^{a+1}\right)^{\frac{1}{3}}\) - step2: Multiply the exponents: \(\sqrt{a+1}\times 8^{\left(a+1\right)\times \frac{1}{3}}\) - step3: Multiply the terms: \(\sqrt{a+1}\times 8^{\frac{1}{3}a+\frac{1}{3}}\) - step4: Rewrite the expression: \(8^{\frac{1}{3}a+\frac{1}{3}}\sqrt{a+1}\) Calculate or simplify the expression \( (7^(n+1) * 14^(n-1)) / (98^(n-1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(7^{n+1}\times 14^{n-1}\right)}{98^{n-1}}\) - step1: Remove the parentheses: \(\frac{7^{n+1}\times 14^{n-1}}{98^{n-1}}\) - step2: Factor the expression: \(\frac{7^{n+1}\times 14^{n-1}}{7^{n-1}\times 14^{n-1}}\) - step3: Reduce the fraction: \(7^{2}\) - step4: Evaluate the power: \(49\) Calculate or simplify the expression \( (2^(x-1) + 2^(x+3)) / (3 * 2^(x+1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2^{x-1}+2^{x+3}\right)}{\left(3\times 2^{x+1}\right)}\) - step1: Remove the parentheses: \(\frac{2^{x-1}+2^{x+3}}{3\times 2^{x+1}}\) - step2: Add the terms: \(\frac{17\times 2^{x-1}}{3\times 2^{x+1}}\) - step3: Reduce the fraction: \(\frac{17}{3\times 2^{2}}\) - step4: Calculate: \(\frac{17}{12}\) Calculate or simplify the expression \( 2.1 * (-6 * x^2)^3 * (-x^(-3))^(-2) / (-3 * x^3)^2 \). Simplify the expression by following steps: - step0: Solution: \(\frac{2.1\left(-6x^{2}\right)^{3}\left(-x^{-3}\right)^{-2}}{\left(-3x^{3}\right)^{2}}\) - step1: Multiply the exponents: \(\frac{2.1\left(-6x^{2}\right)^{3}x^{-3\left(-2\right)}}{\left(-3x^{3}\right)^{2}}\) - step2: Multiply the numbers: \(\frac{2.1\left(-6x^{2}\right)^{3}x^{6}}{\left(-3x^{3}\right)^{2}}\) - step3: Factor the expression: \(\frac{2.1\left(-24\right)\times 9x^{6}\times x^{6}}{9x^{6}}\) - step4: Reduce the fraction: \(2.1\left(-24\right)x^{6}\) - step5: Rewrite the expression: \(-2.1\times 24x^{6}\) - step6: Multiply the terms: \(-50.4x^{6}\) - step7: Rewrite the expression: \(-\frac{252}{5}x^{6}\) - step8: Rewrite the expression: \(-\frac{252x^{6}}{5}\) Calculate or simplify the expression \( (25^(n-1) * 6^n) / (10^(n-1) * 15^n) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(25^{n-1}\times 6^{n}\right)}{\left(10^{n-1}\times 15^{n}\right)}\) - step1: Remove the parentheses: \(\frac{25^{n-1}\times 6^{n}}{10^{n-1}\times 15^{n}}\) - step2: Factor the expression: \(\frac{5^{2n-2}\times 6^{n}}{5^{n-1}\times 2^{n-1}\times 15^{n}}\) - step3: Reduce the fraction: \(\frac{5^{n-1}\times 6^{n}}{2^{n-1}\times 15^{n}}\) - step4: Factor the expression: \(\frac{5^{n-1}\times 6^{n}}{2^{n-1}\times 5^{n}\times 3^{n}}\) - step5: Reduce the fraction: \(\frac{6^{n}}{2^{n-1}\times 5\times 3^{n}}\) - step6: Factor the expression: \(\frac{2^{n}\times 3^{n}}{2^{n-1}\times 5\times 3^{n}}\) - step7: Reduce the fraction: \(\frac{2}{5}\) Calculate or simplify the expression \( (5^(x+2) - 4 * 5^x) / (5^(x-1) + 2 * 5^(x+1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(5^{x+2}-4\times 5^{x}\right)}{\left(5^{x-1}+2\times 5^{x+1}\right)}\) - step1: Remove the parentheses: \(\frac{5^{x+2}-4\times 5^{x}}{5^{x-1}+2\times 5^{x+1}}\) - step2: Subtract the terms: \(\frac{21\times 5^{x}}{5^{x-1}+2\times 5^{x+1}}\) - step3: Add the terms: \(\frac{21\times 5^{x}}{51\times 5^{x-1}}\) - step4: Reduce the fraction: \(\frac{7\times 5}{17}\) - step5: Calculate: \(\frac{35}{17}\) Calculate or simplify the expression \( 2.23 * 0.5^x + 6 * 0.5^x - 4 / (5^(-x)) \). Simplify the expression by following steps: - step0: Solution: \(2.23\times 0.5^{x}+6\times 0.5^{x}-\frac{4}{5^{-x}}\) - step1: Convert the expressions: \(2.23\left(\frac{1}{2}\right)^{x}+6\times 0.5^{x}-\frac{4}{5^{-x}}\) - step2: Convert the expressions: \(2.23\left(\frac{1}{2}\right)^{x}+6\left(\frac{1}{2}\right)^{x}-\frac{4}{5^{-x}}\) - step3: Divide the terms: \(2.23\left(\frac{1}{2}\right)^{x}+6\left(\frac{1}{2}\right)^{x}-4\times 5^{x}\) - step4: Multiply the terms: \(2.23\times 2^{-x}+6\left(\frac{1}{2}\right)^{x}-4\times 5^{x}\) - step5: Multiply the terms: \(2.23\times 2^{-x}+6\times 2^{-x}-4\times 5^{x}\) - step6: Add the terms: \(\frac{823}{25\times 2^{2+x}}-4\times 5^{x}\) - step7: Reduce fractions to a common denominator: \(\frac{823}{25\times 2^{2+x}}-\frac{4\times 5^{x}\times 25\times 2^{2+x}}{25\times 2^{2+x}}\) - step8: Transform the expression: \(\frac{823-4\times 5^{x}\times 25\times 2^{2+x}}{25\times 2^{2+x}}\) - step9: Multiply the terms: \(\frac{823-100\times 5^{x}\times 2^{2+x}}{25\times 2^{2+x}}\) Let's solve each of the problems step by step: ### Problem 2.1 **Expression:** \[ 2.1 \times (-6x^2)^3 \times (-x^{-3})^{-2} \div (-3x^3)^2 \] **Solution:** 1. Calculate \((-6x^2)^3\): \[ (-6)^3 \times (x^2)^3 = -216x^6 \] 2. Calculate \((-x^{-3})^{-2}\): \[ (-1)^{-2} \times (x^{-3})^{-2} = 1 \times x^6 = x^6 \] 3. Calculate \((-3x^3)^2\): \[ (-3)^2 \times (x^3)^2 = 9x^6 \] 4. Combine everything: \[ 2.1 \times (-216x^6) \times (x^6) \div (9x^6) = 2.1 \times -216 \div 9 \times x^{12-6} = -\frac{252x^6}{5} \] ### Problem 2.2 **Expression:** \[ 2.23 \cdot 0.5^x + 6 \cdot 0.5^x - \frac{4}{5^{-x}} \] **Solution:** 1. Combine the first two terms: \[ (2.23 + 6) \cdot 0.5^x = 8.23 \cdot 0.5^x \] 2. Rewrite \(\frac{4}{5^{-x}}\) as \(4 \cdot 5^x\): \[ 8.23 \cdot 0.5^x - 4 \cdot 5^x \] 3. The final expression is: \[ \frac{823 - 100 \cdot 5^x \cdot 2^{2+x}}{25 \cdot 2^{2+x}} \] ### Problem 2.3 **Expression:** \[ \frac{7^{n+1} \cdot 14^{n-1}}{98^{n-1}} \] **Solution:** 1. Rewrite \(14\) and \(98\): \[ 14 = 2 \cdot 7 \quad \text{and} \quad 98 = 14 \cdot 7 = 2 \cdot 7^2 \] 2. Substitute: \[ \frac{7^{n+1} \cdot (2 \cdot 7)^{n-1}}{(2 \cdot 7^2)^{n-1}} = \frac{7^{n+1} \cdot 2^{n-1} \cdot 7^{n-1}}{2^{n-1} \cdot 7^{2(n-1)}} \] 3. Simplify: \[ = \frac{7^{2n}}{7^{2n-2}} = 49 \] ### Problem 2.4 **Expression:** \[ \frac{25^{n-1} \cdot 6^n}{10^{n-1} \cdot 15^n} \] **Solution:** 1. Rewrite \(10\) and \(15\): \[ 10 = 2 \cdot 5 \quad \text{and} \quad 15 = 3 \cdot 5 \] 2. Substitute: \[ = \frac{25^{n-1} \cdot 6^n}{(2 \cdot 5)^{n-1} \cdot (3 \cdot 5)^n} \] 3. Simplify: \[ = \frac{25^{n-1} \cdot 6^n}{2^{n-1} \cdot 5^{n-1} \cdot 3^n \cdot 5^n} = \frac{25^{n-1} \cdot 6^n}{2^{n-1} \cdot 3^n \cdot 5^{2n-1}} \] 4. The final result is: \[ \frac{2}{5} \quad \text{or} \quad 0.4 \] ### Problem 2.5 **Expression:** \[ \frac{2^{x-1} + 2^{x+3}}{3 \cdot 2^{x+1}} \] **Solution:** 1. Factor out \(2^{x-1}\): \[ = \frac{2^{x-1}(1 + 2^4)}{3 \cdot 2^{x+1}} = \frac{2^{x-1} \cdot 17}{3 \cdot 2^{x+1}} \] 2. Simplify: \[ = \frac{17}{3 \cdot 2^2} = \frac{17}{12} \] ### Problem 2.6 **Expression:** \[ \frac{5^{x+2} - 4 \cdot 5^x}{5^{x-1} + 2 \cdot 5^{x+1}} \] **Solution:** 1. Factor out \(5^x\) from the numerator: \[ = \frac{5^x(5^2 - 4)}{5^{x-1} + 2 \cdot 5^{x+1}} = \frac{5^x(25 - 4)}{5^{x-1} + 2 \cdot 5^{x+1}} = \frac{21 \cdot 5^x}{5^{x-1} + 2 \cdot 5^{x+1}} \] 2. Simplify the denominator: \[ = \frac{21 \cdot 5^x}{5^{x-1}(1 + 2 \cdot 5^2)} = \frac{21}{\frac{1 + 50}{5}} = \frac{35}{17} \] ### Problem 2.7 **Expression:** \[ \frac{5^{1000} - 5^{502} + 24}{5^{501} - 5} \] **Solution:** 1. Factor out \(5^{501}\) from the numerator: \[ = \frac{5^{501}(5^{499} - 1) + 24}{5^{501} - 5} \] 2. The expression