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Mathematics

If 25x+1=1255x25^{x+1} = \dfrac{125}{5^x} ; find the value of x .

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25x+1=1255x⇒52(x+1)=535x⇒52x+2=5(3−x)⇒2x+2=3−x⇒2x+x=3−2⇒3x=1⇒x=1325^{x+1} = \dfrac{125}{5^x}\\[1em] ⇒ 5^{2(x+1)} = \dfrac{5^3}{5^x}\\[1em] ⇒ 5^{2x + 2} = 5^{(3 - x)}\\[1em] ⇒ 2x + 2 = 3 - x\\[1em] ⇒ 2x + x = 3 - 2\\[1em] ⇒ 3x = 1\\[1em] ⇒ x = \dfrac{1}{3}

Hence, the value of x = 13\dfrac{1}{3}.

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