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Mathematics

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{(625−12)−14}2=(0.2)4−3x\Big{\Big(625^{-\dfrac{1}{2}}\Big)^{-\dfrac{1}{4}}\Big}^2 = (0.2)^{4-3x}

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Answer

{(625−12)−14}2=(0.2)4−3x⇒{(625−12)−24}=(0.2)4−3x⇒{(625−12)−12}=(0.2)4−3x⇒{(625−12×−12)}=(0.2)4−3x⇒62514=(0.2)4−3x⇒(54)14=(0.2)4−3x⇒5=(0.2)4−3x⇒5=2104−3x⇒5=154−3x⇒5=53x−4⇒3x−4=1⇒3x=1+4⇒3x=5⇒x=53=123\Big{\Big(625^{-\dfrac{1}{2}}\Big)^{-\dfrac{1}{4}}\Big}^2 = (0.2)^{4-3x}\\[1em] \Rightarrow \Big{\Big(625^{-\dfrac{1}{2}}\Big)^{-\dfrac{2}{4}}\Big} = (0.2)^{4-3x}\\[1em] \Rightarrow \Big{\Big(625^{-\dfrac{1}{2}}\Big)^{-\dfrac{1}{2}}\Big} = (0.2)^{4-3x}\\[1em] \Rightarrow \Big{\Big(625^{-\dfrac{1}{2} \times -\dfrac{1}{2}}\Big)\Big} = (0.2)^{4-3x}\\[1em] \Rightarrow 625^{\dfrac{1}{4}} = (0.2)^{4-3x}\\[1em] \Rightarrow {(5^4)}^{\dfrac{1}{4}} = (0.2)^{4-3x}\\[1em] \Rightarrow 5 = (0.2)^{4-3x}\\[1em] \Rightarrow 5 = \dfrac{2}{10}^{4-3x}\\[1em] \Rightarrow 5 = \dfrac{1}{5}^{4-3x}\\[1em] \Rightarrow 5 = 5^{3x-4}\\[1em] \Rightarrow 3x - 4 = 1\\[1em] \Rightarrow 3x = 1 + 4\\[1em] \Rightarrow 3x = 5\\[1em] \Rightarrow x = \dfrac{5}{3} = 1\dfrac{2}{3}

Hence, the value of x = 1231\dfrac{2}{3}.

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