If 2x = 3y = 12z, show that
Question:

If $2^{x}=3^{y}=12^{z}$, show that $1 / z=1 / y+2 / x$

Solution:

$2^{x}=3^{y}=(2 \times 3 \times 2)^{z}$

$2^{x}=3^{y}=\left(2^{2} \times 3\right)^{z}$

$2^{x}=3^{y}=\left(2^{2 z} \times 3^{z}\right)$

$2^{x}=3^{y}=12^{z}=k$

$2=k^{1 / x}$

$3=k^{1 / y}$

$12=k^{1 / z}$

$12=2 \times 3 \times 2$

$12=k^{1 / z}=k^{1 / y} \times k^{1 / x} \times k^{1 / x}$

$k^{1 / z}=k^{2 / x}+{ }^{1 / y}$

$1 / z=1 / y+2 / x$

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