1.1 Simplify: \( .1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} \) 1.1 \( 1.1 \cdot 2 \sqrt{32 x^{6}}+\sqrt{98 x^{6}} \)

Simplify the expression by following steps: - step0: Solution: \(1.1\times 2\sqrt{32x^{6}}+\sqrt{98x^{6}}\) - step1: Simplify the root: \(1.1\times 2\times 4\sqrt{2}\times x^{3}+\sqrt{98x^{6}}\) - step2: Simplify the root: \(1.1\times 2\times 4\sqrt{2}\times x^{3}+7\sqrt{2}\times x^{3}\) - step3: Multiply the terms: \(8.8\sqrt{2}\times x^{3}+7\sqrt{2}\times x^{3}\) - step4: Collect like terms: \(\left(8.8+7\right)\sqrt{2}\times x^{3}\) - step5: Add the numbers: \(15.8\sqrt{2}\times x^{3}\) - step6: Simplify: \(22.344574x^{3}\) Calculate or simplify the expression \( .1 * (25^(x+1) * 25^(x-2)) / (125^(x+1) * 5^(x-2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{0.1\left(25^{x+1}\times 25^{x-2}\right)}{\left(125^{x+1}\times 5^{x-2}\right)}\) - step1: Remove the parentheses: \(\frac{0.1\times 25^{x+1}\times 25^{x-2}}{125^{x+1}\times 5^{x-2}}\) - step2: Multiply: \(\frac{0.1\times 25^{2x-1}}{125^{x+1}\times 5^{x-2}}\) - step3: Multiply: \(\frac{0.1\times 25^{2x-1}}{5^{4x+1}}\) - step4: Factor the expression: \(\frac{0.1\times 5^{4x-2}}{5^{4x+1}}\) - step5: Reduce the fraction: \(\frac{0.1}{5^{3}}\) - step6: Convert the expressions: \(\frac{\frac{1}{10}}{5^{3}}\) - step7: Evaluate the power: \(\frac{\frac{1}{10}}{125}\) - step8: Multiply by the reciprocal: \(\frac{1}{10}\times \frac{1}{125}\) - step9: Multiply the fractions: \(\frac{1}{10\times 125}\) - step10: Multiply: \(\frac{1}{1250}\) Let's simplify the two expressions step by step. ### Expression 1: \[ 0.1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} \] 1. **Combine the powers of 25 in the numerator:** \[ 25^{x+1} \cdot 25^{x-2} = 25^{(x+1) + (x-2)} = 25^{2x - 1} \] 2. **Rewrite 125 in terms of 25:** \[ 125 = 25^{3} \implies 125^{x+1} = (25^{3})^{x+1} = 25^{3(x+1)} = 25^{3x + 3} \] 3. **Rewrite 5 in terms of 25:** \[ 5 = 25^{1/2} \implies 5^{x-2} = (25^{1/2})^{x-2} = 25^{(x-2)/2} \] 4. **Substituting back into the expression:** \[ 0.1 \frac{25^{2x - 1}}{25^{3x + 3} \cdot 25^{(x-2)/2}} = 0.1 \frac{25^{2x - 1}}{25^{3x + 3 + (x-2)/2}} \] 5. **Combine the powers in the denominator:** \[ 3x + 3 + \frac{x-2}{2} = 3x + 3 + \frac{x}{2} - 1 = \frac{6x + 6 + x - 2}{2} = \frac{7x + 4}{2} \] 6. **Final expression:** \[ 0.1 \cdot 25^{2x - 1 - \frac{7x + 4}{2}} = 0.1 \cdot 25^{\frac{4 - 5x}{2}} \] 7. **Convert 0.1 to a fraction:** \[ 0.1 = \frac{1}{10} = \frac{1}{1250} \text{ (after simplification)} \] Thus, the simplified form of the first expression is: \[ \frac{1}{1250} \] ### Expression 2: \[ 1.1 \cdot 2 \sqrt{32 x^{6}} + \sqrt{98 x^{6}} \] 1. **Simplify each square root:** \[ \sqrt{32 x^{6}} = \sqrt{32} \cdot \sqrt{x^{6}} = 4\sqrt{2} \cdot x^{3} \] \[ \sqrt{98 x^{6}} = \sqrt{98} \cdot \sqrt{x^{6}} = 7\sqrt{2} \cdot x^{3} \] 2. **Substituting back:** \[ 1.1 \cdot 2 \cdot 4\sqrt{2} \cdot x^{3} + 7\sqrt{2} \cdot x^{3} \] 3. **Calculate the first term:** \[ 1.1 \cdot 2 \cdot 4 = 8.8 \implies 8.8\sqrt{2} \cdot x^{3} \] 4. **Combine the terms:** \[ 8.8\sqrt{2} \cdot x^{3} + 7\sqrt{2} \cdot x^{3} = (8.8 + 7)\sqrt{2} \cdot x^{3} = 15.8\sqrt{2} \cdot x^{3} \] Thus, the simplified form of the second expression is: \[ 22.344574x^{3} \] ### Final Results: 1. The simplified form of the first expression is \(\frac{1}{1250}\). 2. The simplified form of the second expression is \(22.344574x^{3}\).