If `3^(x+y) = 81 and 81 ^(x-y) = 3`, then find the values of `x and y`.

Answer

To solve the equations \(3^{(x+y)} = 81\) and \(81^{(x-y)} = 3\), we can follow these steps:

### Step 1: Rewrite the equations using the same base
We know that \(81\) can be expressed as a power of \(3\). Specifically, \(81 = 3^4\).

So, we can rewrite the first equation:
\[ 3^{(x+y)} = 3^4 \]

### Step 2: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ x + y = 4 \quad \text{(Equation 1)} \]

### Step 3: Rewrite the second equation
Next, we rewrite the second equation. Since \(81 = 3^4\), we can express it as:
\[ (3^4)^{(x-y)} = 3 \]

### Step 4: Simplify the second equation
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ 3^{4(x-y)} = 3^1 \]

### Step 5: Set the exponents equal to each other again
Again, since the bases are the same, we can set the exponents equal:
\[ 4(x - y) = 1 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations
Now we have a system of two equations:
1. \(x + y = 4\)
2. \(4(x - y) = 1\)

From Equation 2, we can simplify:
\[ x - y = \frac{1}{4} \quad \text{(Equation 3)} \]

### Step 7: Solve for \(x\) and \(y\)
Now we can solve these two equations (Equation 1 and Equation 3) simultaneously.

From Equation 1:
\[ x + y = 4 \quad \text{(1)} \]
From Equation 3:
\[ x - y = \frac{1}{4} \quad \text{(3)} \]

We can add Equation 1 and Equation 3:
\[ (x + y) + (x - y) = 4 + \frac{1}{4} \]
This simplifies to:
\[ 2x = 4 + \frac{1}{4} \]
Converting \(4\) to a fraction:
\[ 2x = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]
Now, divide by \(2\):
\[ x = \frac{17}{8} \]

### Step 8: Substitute \(x\) back to find \(y\)
Now substitute \(x\) back into Equation 1 to find \(y\):
\[ \frac{17}{8} + y = 4 \]
Convert \(4\) to a fraction:
\[ \frac{17}{8} + y = \frac{32}{8} \]
Now, solve for \(y\):
\[ y = \frac{32}{8} - \frac{17}{8} = \frac{15}{8} \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = \frac{17}{8}, \quad y = \frac{15}{8} \]