The value of y in the solution of
`2 ^(x + y) = 2 ^( x - y) = 16` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \(2^{(x + y)} = 2^{(x - y)} = 16\), we will follow these steps:

### Step 1: Rewrite the Equation
Since both expressions are equal to 16, we can rewrite them as:
1. \(2^{(x + y)} = 16\)
2. \(2^{(x - y)} = 16\)

### Step 2: Convert 16 to Base 2
We know that \(16\) can be expressed as a power of \(2\):
\[ 16 = 2^4 \]
Thus, we can rewrite our equations as:
1. \(2^{(x + y)} = 2^4\)
2. \(2^{(x - y)} = 2^4\)

### Step 3: Equate the Exponents
Since the bases are the same, we can equate the exponents:
1. \(x + y = 4\)
2. \(x - y = 4\)

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \(x + y = 4\) (Equation 1)
2. \(x - y = 4\) (Equation 2)

We can solve these equations by adding them together:
\[ (x + y) + (x - y) = 4 + 4 \]
This simplifies to:
\[ 2x = 8 \]
Dividing both sides by 2 gives:
\[ x = 4 \]

### Step 5: Substitute to Find y
Now that we have the value of \(x\), we can substitute it back into either equation to find \(y\). Using Equation 1:
\[ 4 + y = 4 \]
Subtracting 4 from both sides gives:
\[ y = 0 \]

### Final Answer
The value of \(y\) is:
\[ \boxed{0} \]

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