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Question : 17 of 86

Marks: +1, -0

If

x^{\frac{1}{p}}=y^{\frac{1}{q}}=z^{\frac{1}{r}}

x y z = 1

p+q+r

is

1
0
$\frac{1}{2}$
none

Solution:

Concept:

Express x, y, z in terms of a common base and use the product condition to find the sum of exponents.

Explanation:

Let

x^{1/p} = y^{1/q} = z^{1/r} = k

.

Then

x = k^p

,

y = k^q

,

z = k^r

.

Given

xyz = 1

, so

k^p \cdot k^q \cdot k^r = k^{p+q+r} = 1

.

For a positive

k \neq 1

, the only way

k^{p+q+r}=1

is when

p+q+r=0

.

Even if

k=1

, the condition holds for any

p,q,r

, but the typical problem assumes a general case, so

p+q+r=0

.

Answer:

0

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