Question

if$2^{x} = 3^{y} = 6^{z}$show that$\frac{1}{z} = \frac{1}{y} + \frac{2}{x}$

Answer 1

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Given Question is

2^x = 3^y = 6^z

Let 2^x = 3^y = 6^z = k

2^x = k , 3^y = k And 6^z = k

2 = k^(1/x) , 3 = k^(1/y) And 6 = k^(1/z)

6 = k^(1/z)

3 × 2 = k^(1/z)

k^(1/x) × k^(1/y) = k^(1/z)

k^{(1/x) + (1/y)} = k^(1/z)

Now, Compare powers of k we have

1/x + 1/y = 1/z Hence, proved:

Note:-

IF xⁿ = y

Then, x = y^(1/n)

Answer 2

answer refer to the attachment