1.1 Simplify: \( .1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} \) 1.1 \( 1.1 \cdot 2 \sqrt{32 x^{6}}+\sqrt{98 x^{6}} \)
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To simplify the expression \( .1 \frac{25^{x+1} \cdot 25^{x-2}}{125^{x+1} \cdot 5^{x-2}} \), we start by expressing everything in terms of base 5. We have \( 25 = 5^2 \) and \( 125 = 5^3 \). Therefore: \[ .1 \frac{(5^2)^{x+1} \cdot (5^2)^{x-2}}{(5^3)^{x+1} \cdot 5^{x-2}} = .1 \frac{5^{2(x+1)+2(x-2)}}{5^{3(x+1)} \cdot 5^{x-2}} = .1 \frac{5^{2x + 2 + 2x - 4}}{5^{3x + 3 + x - 2}} = .1 \frac{5^{4x - 2}}{5^{4x + 1}} = .1 \cdot 5^{-3} = \frac{.1}{125} = \frac{1}{1250}. \] Now simplifying \( 1.1 \cdot 2 \sqrt{32 x^{6}}+\sqrt{98 x^{6}} \): First, simplify the square roots: \(\sqrt{32x^6} = \sqrt{16 \cdot 2 \cdot x^6} = 4x^3\sqrt{2}\) and \(\sqrt{98x^6} = \sqrt{49 \cdot 2 \cdot x^6} = 7x^3\sqrt{2}\). Now substitute these back into the expression: \[ 1.1 \cdot 2(4x^3\sqrt{2}) + 7x^3\sqrt{2} = 8.8x^3\sqrt{2} + 7x^3\sqrt{2} = (8.8 + 7)x^3\sqrt{2} = 15.8x^3\sqrt{2}. \] So, the overall simplified expression is: \[ \frac{1}{1250} \quad \text{and} \quad 15.8x^3\sqrt{2}. \]
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