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, we start with the given equations:

1. \( a^{1/m} = b^{1/n} = c^{1/p} \)
2. \( abc = 1 \)

Let’s denote the common value of \( a^{1/m} \), \( b^{1/n} \), and \( c^{1/p} \) as \( k \). Thus, we can write:

\[ a^{1/m} = k \implies a = k^m \]
\[ b^{1/n} = k \implies b = k^n \]
\[ c^{1/p} = k \implies c = k^p \]

Now, substituting these expressions for \( a \), \( b \), and \( c \) into the equation \( abc = 1 \):

\[ abc = (k^m)(k^n)(k^p) = k^{m+n+p} \]

Since we know \( abc = 1 \), we can set this equal to \( 1 \):

\[ k^{m+n+p} = 1 \]

The equation \( k^{m+n+p} = 1 \) holds true if \( m+n+p = 0 \) (assuming \( k \neq 0 \)). This is because any non-zero number raised to the power of 0 equals 1.

Thus, we conclude:

\[ m + n + p = 0 \]

### Final Answer:
\[ m + n + p = 0 \]