If `a^(x)=(x+y+z)^(y), a^(y)=(x+y+z)^(z), a^(z)=(x+y+z)^(x)`, then :

Options

A. `3(x+y+z)^(y)=a`
B. `2a=x+y+z`
C. `x+y+z=0`
D. `x+y+z=a`

Answer

To solve the given problem, we need to find the value of \(a\) given the equations:

\[ a^x = (x + y + z)^y \]
\[ a^y = (x + y + z)^z \]
\[ a^z = (x + y + z)^x \]

Let's proceed step-by-step:

### Step 1: Multiply the three given equations
Multiply the left-hand sides and the right-hand sides of the three equations:

\[ (a^x) \cdot (a^y) \cdot (a^z) = ( (x + y + z)^y ) \cdot ( (x + y + z)^z ) \cdot ( (x + y + z)^x ) \]

### Step 2: Simplify the left-hand side
Using the properties of exponents, we can combine the left-hand side:

\[ a^{x + y + z} \]

### Step 3: Simplify the right-hand side
Similarly, combine the right-hand side using the properties of exponents:

\[ (x + y + z)^{y + z + x} \]

### Step 4: Equate the simplified expressions
Now we have:

\[ a^{x + y + z} = (x + y + z)^{x + y + z} \]

### Step 5: Compare the exponents
Since the bases are the same, we can equate the exponents:

\[ a = x + y + z \]

Thus, the value of \(a\) is:

\[ \boxed{x + y + z} \]