#### Question

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

Question

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

Log both sides of 2^{x+y} = 125,

x+y = \log_{2}125 = \log_{2}5^3 = 3\log_{2}5

(1)

Similarly

x - y = \log_{2}25 = \log_{2}5^2 = 2\log_{2}5

(2)

Addition of equations (1) and (2) gives

2x = 3\log_{2}5+ +2\log_{2}5 = 5\log_{2}5

(3)

Subtracting (2) from (1)

2y = \log_{2}5

(4)

Dividing (3) by (4) gives

\dfrac{x}{y} = 5

Therefore,

2^{\frac{x}{y} } = 2^5 = 32