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Question Number 59730 by muneshkumar last updated on 14/May/19

5^(2x−1 ) =25^(x−1) +100 find value of 3^(3−x)

$$\mathrm{5}^{\mathrm{2}{x}−\mathrm{1}\:} =\mathrm{25}^{{x}−\mathrm{1}} +\mathrm{100}\:{find}\:{value}\:{of}\:\mathrm{3}^{\mathrm{3}−{x}} \\ $$

Answered by tanmay last updated on 14/May/19

(5^(2x) /5)=(5^2 )^(x−1) +100 (((5^x )^2 )/5)=(5^(2x) /5^2 )+100 5^x =a (a^2 /5)=(a^2 /(25))+100 (a^2 /5)−(a^2 /(25))=100 ((5a^2 −a^2 )/(25))=100 4a^2 =100×25 a^2 =(((10×5)/2))^2 =5^4 5^(2x) =5^4 2x=4 x=2 so 3^(3−2) 3^(3−2) 3^1 3

$$\frac{\mathrm{5}^{\mathrm{2}{x}} }{\mathrm{5}}=\left(\mathrm{5}^{\mathrm{2}} \right)^{{x}−\mathrm{1}} +\mathrm{100} \\ $$$$\frac{\left(\mathrm{5}^{{x}} \right)^{\mathrm{2}} }{\mathrm{5}}=\frac{\mathrm{5}^{\mathrm{2}{x}} }{\mathrm{5}^{\mathrm{2}} }+\mathrm{100} \\ $$$$\mathrm{5}^{{x}} ={a} \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{5}}=\frac{{a}^{\mathrm{2}} }{\mathrm{25}}+\mathrm{100} \\ $$$$\frac{{a}^{\mathrm{2}} }{\mathrm{5}}−\frac{{a}^{\mathrm{2}} }{\mathrm{25}}=\mathrm{100} \\ $$$$\frac{\mathrm{5}{a}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{25}}=\mathrm{100} \\ $$$$\mathrm{4}{a}^{\mathrm{2}} =\mathrm{100}×\mathrm{25} \\ $$$${a}^{\mathrm{2}} =\left(\frac{\mathrm{10}×\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{4}} \\ $$$$\mathrm{5}^{\mathrm{2}{x}} =\mathrm{5}^{\mathrm{4}} \\ $$$$\mathrm{2}{x}=\mathrm{4} \\ $$$${x}=\mathrm{2} \\ $$$${so} \\ $$$$\mathrm{3}^{\mathrm{3}−\mathrm{2}} \\ $$$$\mathrm{3}^{\mathrm{3}−\mathrm{2}} \\ $$$$\mathrm{3}^{\mathrm{1}} \\ $$$$\mathrm{3} \\ $$$$ \\ $$$$ \\ $$

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