$$2^{x+y} = 125, 2^{x-y} = 25$$, find the value of $$2^{\frac{x}{y} }$$

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$$2^{x+y} = 125, 2^{x-y} = 25$$, find the value of $$2^{\frac{x}{y}  }$$

Uniteasy Admin · Accepted Answer

Log both sides of $$2^{x+y} = 125$$,
$$  x+y = \log_{2}125 = \log_{2}5^3  = 3\log_{2}5 $$n 
Similarly
$$x - y = \log_{2}25 = \log_{2}5^2  = 2\log_{2}5$$n
Addition of equations (1) and (2) gives
$$2x = 3\log_{2}5+ +2\log_{2}5 = 5\log_{2}5$$n
Subtracting (2) from (1)
$$2y = \log_{2}5$$n
Dividing (3) by (4) gives
$$\dfrac{x}{y} = 5 $$
Therefore,
$$2^{\frac{x}{y} } = 2^5 = 32$$

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