Simplify the following:
52(x+6)×(25)−7+2x(125)2x\dfrac{5^{2(x + 6)} \times (25)^{-7 + 2x}}{(125)^{2x}}(125)2x52(x+6)×(25)−7+2x
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Given,
⇒52(x+6)×(25)−7+2x(125)2x⇒52x+12×(52)−7+2x(53)2x⇒52x+12×5−14+4x56x⇒52x+12+(−14+4x)56x⇒56x−2.5−6x⇒56x−2+(−6x)⇒5−2=152=125.\Rightarrow \dfrac{5^{2(x + 6)} \times (25)^{-7 + 2x}}{(125)^{2x}} \\[1em] \Rightarrow \dfrac{5^{2x + 12} \times (5^2)^{-7 + 2x}}{(5^3)^{2x}} \\[1em] \Rightarrow \dfrac{5^{2x + 12} \times 5^{-14 + 4x}}{5^{6x}} \\[1em] \Rightarrow \dfrac{5^{2x + 12 + (-14 + 4x)}}{5^{6x}} \\[1em] \Rightarrow 5^{6x - 2}.5^{-6x} \\[1em] \Rightarrow 5^{6x - 2 + (-6x)} \\[1em] \Rightarrow 5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}.⇒(125)2x52(x+6)×(25)−7+2x⇒(53)2x52x+12×(52)−7+2x⇒56x52x+12×5−14+4x⇒56x52x+12+(−14+4x)⇒56x−2.5−6x⇒56x−2+(−6x)⇒5−2=521=251.
Hence, 52(x+6)×(25)−7+2x(125)2x=125\dfrac{5^{2(x + 6)} \times (25)^{-7 + 2x}}{(125)^{2x}} = \dfrac{1}{25}(125)2x52(x+6)×(25)−7+2x=251.
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