If `a^(1//m) = b^(1//n) = c^(1//p)` and abc = 1 then `m + n + p` is equal to :

Options

A. 0
B. 2
C. 1
D. `-2`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Set up the equation
We are given that:
\[ a^{\frac{1}{m}} = b^{\frac{1}{n}} = c^{\frac{1}{p}} \]
Let's denote this common value as \( k \). Therefore, we can write:
\[ a^{\frac{1}{m}} = k, \quad b^{\frac{1}{n}} = k, \quad c^{\frac{1}{p}} = k \]

### Step 2: Express \( a, b, c \) in terms of \( k \)
From the equations above, we can express \( a, b, c \) as:
\[ a = k^m, \quad b = k^n, \quad c = k^p \]

### Step 3: Use the condition \( abc = 1 \)
We know that:
\[ abc = 1 \]
Substituting the values of \( a, b, c \) from Step 2, we get:
\[ (k^m)(k^n)(k^p) = 1 \]

### Step 4: Combine the powers
Using the property of exponents, we can combine the left-hand side:
\[ k^{m+n+p} = 1 \]

### Step 5: Rewrite 1 as a power of \( k \)
We can express 1 as:
\[ 1 = k^0 \]
Thus, we have:
\[ k^{m+n+p} = k^0 \]

### Step 6: Compare the exponents
Since the bases are the same (assuming \( k \neq 0 \)), we can equate the exponents:
\[ m + n + p = 0 \]

### Conclusion
Thus, the value of \( m + n + p \) is:
\[ \boxed{0} \]

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