If x to the power 1/p=y to the power 1/q=z to the power 1/r and xyx=1 then find value of p q r?

Question

Answer

Solution:

Given:

x^(1/p) = y^(1/q) = z^(1/r)

xyz = 1

Let us assume:

k = x^(1/p) = y^(1/q) = z^(1/r)

Then k^p = x, k^q = y, and k^r = z

Substituting these values in the equation xyz=1, we get:

k^(p+q+r) = 1

Taking the logarithm of both sides, we get:

(p+q+r) log(k) = 0

Since k is positive, log(k) is also positive. Therefore, (p+q+r) must be equal to zero.

Hence, p+q+r=0.

We also know that k = x^(1/p) = y^(1/q) = z^(1/r). Therefore, we can write:

k^(p+q+r) = x^(q+r) y^(p+r) z^(p+q) = 1

Substituting p+q+r=0, we get:

x^(q+r) y^(p+r) z^(p+q) = x^(-p) y^(-q) z^(-r)

Multiplying both sides by x^p y^q z^r, we get:

x^q y^r z^p = 1

This implies that x, y, and z are the roots of the equation:

t^3 = 1

Therefore, x = y = z = 1.

Substituting this in p+q+r=0, we get:

p+q+r=0

Since x=y=z=1, we have:

k = x^(1/p) = y^(1/q) = z^(1/r) = 1

Therefore, p=q=r=0.

Hence, the values of p,q, and r are 0.

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