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Solve-for-x-in-the-equation-625-x-5-200-x-3-




Question Number 11263 by tawa last updated on 18/Mar/17
Solve for x in the equation .  625^(x − 5)  = 200(√x^3 )
$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{equation}\:. \\ $$$$\mathrm{625}^{\mathrm{x}\:−\:\mathrm{5}} \:=\:\mathrm{200}\sqrt{\mathrm{x}^{\mathrm{3}} } \\ $$
Answered by ajfour last updated on 18/Mar/17
5^(4(x−5))  = 200x^(3/2)   5^(4x−20)  = 5^2 (4x)^(3/2)   5^(4x−22)  = (4x)^(3/2)   5^(4x)  (4x)^(−3/2)  = 5^(22)   clearly if 4x = 25  5^(25) (5)^(−3)  = 5^(22)   Hence 4x=25      x =((25)/4)
$$\mathrm{5}^{\mathrm{4}\left(\mathrm{x}−\mathrm{5}\right)} \:=\:\mathrm{200x}^{\mathrm{3}/\mathrm{2}} \\ $$$$\mathrm{5}^{\mathrm{4x}−\mathrm{20}} \:=\:\mathrm{5}^{\mathrm{2}} \left(\mathrm{4x}\right)^{\mathrm{3}/\mathrm{2}} \\ $$$$\mathrm{5}^{\mathrm{4x}−\mathrm{22}} \:=\:\left(\mathrm{4x}\right)^{\mathrm{3}/\mathrm{2}} \\ $$$$\mathrm{5}^{\mathrm{4x}} \:\left(\mathrm{4x}\right)^{−\mathrm{3}/\mathrm{2}} \:=\:\mathrm{5}^{\mathrm{22}} \\ $$$$\mathrm{clearly}\:\mathrm{if}\:\mathrm{4x}\:=\:\mathrm{25} \\ $$$$\mathrm{5}^{\mathrm{25}} \left(\mathrm{5}\right)^{−\mathrm{3}} \:=\:\mathrm{5}^{\mathrm{22}} \\ $$$$\mathrm{Hence}\:\mathrm{4x}=\mathrm{25} \\ $$$$\:\:\:\:\mathrm{x}\:=\frac{\mathrm{25}}{\mathrm{4}}\:\: \\ $$

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