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x-y-positive-real-numbers-If-15-x-9-y-Find-5-3x-2y-x-




Question Number 199230 by hardmath last updated on 29/Oct/23
x , y :   positive real numbers  If :   15^x  = 9^y   Find:   5^((3x)/(2y − x))   =  ?
$$\mathrm{x}\:,\:\mathrm{y}\::\:\:\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{If}\::\:\:\:\mathrm{15}^{\boldsymbol{\mathrm{x}}} \:=\:\mathrm{9}^{\boldsymbol{\mathrm{y}}} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{5}^{\frac{\mathrm{3}\boldsymbol{\mathrm{x}}}{\mathrm{2}\boldsymbol{\mathrm{y}}\:−\:\boldsymbol{\mathrm{x}}}} \:\:=\:\:? \\ $$
Answered by AST last updated on 29/Oct/23
3^x 5^x =3^(2y) ⇒5^x =3^(2y−x) ⇒5^((3x)/(2y−x)) =3^3 =27
$$\mathrm{3}^{{x}} \mathrm{5}^{{x}} =\mathrm{3}^{\mathrm{2}{y}} \Rightarrow\mathrm{5}^{{x}} =\mathrm{3}^{\mathrm{2}{y}−{x}} \Rightarrow\mathrm{5}^{\frac{\mathrm{3}{x}}{\mathrm{2}{y}−{x}}} =\mathrm{3}^{\mathrm{3}} =\mathrm{27} \\ $$

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