If `a^(x)=b,b^(y)=candxyz=2`, then `c^(z)=`?

Options

A. a
B. b
C. `a^(2)`
D. 1

Answer

To solve the problem step by step, we need to find the value of \( c^z \) given the equations \( a^x = b \), \( b^y = c \), and \( xyz = 2 \).

### Step 1: Substitute the values of \( x \), \( y \), and \( z \)
We know from the problem statement that \( x = 2 \), \( y = 2 \), and \( z = 2 \).

### Step 2: Express \( b \) in terms of \( a \)
From the equation \( a^x = b \), substituting \( x = 2 \):
\[ a^2 = b \]

### Step 3: Express \( c \) in terms of \( b \)
From the equation \( b^y = c \), substituting \( y = 2 \):
\[ b^2 = c \]

### Step 4: Substitute \( b \) into the equation for \( c \)
Now, we substitute \( b = a^2 \) into \( c = b^2 \):
\[ c = (a^2)^2 = a^{2 \cdot 2} = a^4 \]

### Step 5: Find \( c^z \)
Now we need to find \( c^z \). Substituting \( c = a^4 \) and \( z = 2 \):
\[ c^z = (a^4)^2 = a^{4 \cdot 2} = a^8 \]

### Final Answer
Thus, the value of \( c^z \) is:
\[ c^z = a^8 \]

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