Question

75^1/x = 45^1/y = 15^1/z.
Show that x + y = 3z

Answer 1

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Answer 2

Given :  $\sqrt[x]{75} =\sqrt[y]{45} =\sqrt[z]{15}$

To Find : which of the statement is true ...

1. x+y=2z

2.x+y=3z

3.x-y=2z

4.x-y=3z

Solution:

$\sqrt[x]{75} =\sqrt[y]{45} =\sqrt[z]{15}$

$\implies \sqrt[x]{5\times 5\times3} =\sqrt[y]{3\times 3\times5} =\sqrt[z]{3\times 5}$

$\implies (5\times 5\times3)^{\frac{1}{x}} = (3\times 3\times5)^{\frac{1}{y}} = (3\times 5 )^{\frac{1}{z}}$

$\implies (5)^{\frac{2}{x}}(3)^{\frac{1}{x}} = (3)^{\frac{2}{y}} (5)^{\frac{1}{y}} = (3 )^{\frac{1}{z}} (5 )^{\frac{1}{z}}$

Equating 1st two terms

$(5)^{\frac{2}{x}}(3)^{\frac{1}{x}} = (3)^{\frac{2}{y}} (5)^{\frac{1}{y}}$

$(5)^{\frac{2}{x}-\frac{1}{y} } = (3)^{\frac{2}{y}-\frac{1}{x}}$

$(5)^{\frac{2y-x}{xy} } = (3)^{\frac{2x-y}{xy}}$

$(5)^{\frac{2y-x}{2x-y} } = 3$   Eq1

Similarly Equating 1st & 3 term

$(5)^{\frac{2z-x}{x-z} } = 3$    Eq2

Equating Eq1 and Eq2 and Equating powers of 5

( 2y - x ) /(2 x - y)  =  (2z - x)/(x - z)

=> 2xy - x² - 2yz + xz  = 4xz - 2x² - 2yz  + xy

=> x²  + xy   = 3xz

=> x + y = 3z

QED

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