Question

If a,b,x, y and z are numbers greater than 1 and a^x=b^y=(ab)^z and 1/x + 1/y = k/z then the value of k is :

Answer 1

SOLUTION

GIVEN

If a,b,x, y and z are numbers greater than 1 and

$\displaystyle \sf{ {a}^{x} = {b}^{y} = {(ab)}^{z} \: \: and \: \frac{1}{x} + \frac{1}{y} = \frac{k}{z} }$

TO DETERMINE

The value of k

EVALUATION

Let

$\displaystyle \sf{ {a}^{x} = {b}^{y} = {(ab)}^{z} = p }$

So that we get

$\displaystyle \sf{ a = {p}^{ \frac{1}{x} } }$

$\displaystyle \sf{ b = {p}^{ \frac{1}{y} } }$

$\displaystyle \sf{ ab = {p}^{ \frac{1}{z} } }$

Now

$\displaystyle \sf{ a \times b = ab }$

$\displaystyle \sf{ \implies \: {p}^{ \frac{1}{x} } \times {p}^{ \frac{1}{y} } = { p}^{ \frac{1}{z} } }$

$\displaystyle \sf{ \implies \: {p}^{ \frac{1}{x} + \frac{1}{y} } = { p}^{ \frac{1}{z} } }$

$\displaystyle \sf{ \implies \: \frac{1}{x} + \frac{1}{y} = \frac{1}{z} }$

Now Comparing with

$\displaystyle \sf{ \implies \: \frac{1}{x} + \frac{1}{y} = \frac{k}{z} }$

We get k = 1

FINAL ANSWER

Hence the required value of k = 1

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