If `x^(1//p)=y^(1//q)=z^(1//r) and xyz =1, " then the value of " p+q+r` would be

Options

A. 0
B. 1
C. 2
D. `a+x+z`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
\[ x^{\frac{1}{p}} = y^{\frac{1}{q}} = z^{\frac{1}{r}} \quad \text{and} \quad xyz = 1 \]

2. **Let’s Introduce a Variable**:
Let \( k = x^{\frac{1}{p}} = y^{\frac{1}{q}} = z^{\frac{1}{r}} \).

3. **Express \( x, y, z \) in terms of \( k \)**:
- From \( x^{\frac{1}{p}} = k \), we can express \( x \) as:
\[ x = k^p \]
- From \( y^{\frac{1}{q}} = k \), we can express \( y \) as:
\[ y = k^q \]
- From \( z^{\frac{1}{r}} = k \), we can express \( z \) as:
\[ z = k^r \]

4. **Substituting into the Product**:
Now, substituting \( x, y, z \) into the equation \( xyz = 1 \):
\[ xyz = (k^p)(k^q)(k^r) = k^{p+q+r} \]

5. **Setting the Equation**:
Since we know \( xyz = 1 \), we can set up the equation:
\[ k^{p+q+r} = 1 \]

6. **Analyzing the Equation**:
The only way for \( k^{p+q+r} \) to equal 1 is if the exponent \( p + q + r \) is equal to 0 (assuming \( k \) is not zero):
\[ p + q + r = 0 \]

7. **Final Result**:
Therefore, the value of \( p + q + r \) is:
\[ \boxed{0} \]