If `a^(x+y) =a^(6) and x gt y" by "2` find x.

A. 1

B. 2

C. 3

D. 4

Options

A. D
B. C
C. A
D. B

Answer

To solve the equation \( a^{(x+y)} = a^{6} \) given that \( x > y \) by 2, we can follow these steps:

### Step 1: Set up the equations
From the equation \( a^{(x+y)} = a^{6} \), since the bases are the same (assuming \( a \neq 0 \) and \( a \neq 1 \)), we can equate the exponents:
\[ x + y = 6 \]

### Step 2: Express the second condition
We are also given that \( x \) is greater than \( y \) by 2. This can be expressed as:
\[ x - y = 2 \]

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( x + y = 6 \)
2. \( x - y = 2 \)

We can solve these equations simultaneously.

### Step 4: Add the two equations
Adding the two equations will help eliminate \( y \):
\[ (x + y) + (x - y) = 6 + 2 \]
This simplifies to:
\[ 2x = 8 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 2:
\[ x = 4 \]

### Step 6: Conclusion
Thus, the value of \( x \) is \( 4 \).

### Final Answer
The answer is \( D. 4 \).

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