The sum of values of `x` satisfying the equation `(31+8sqrt(15))^x^(2-3)+1=(32+8sqrt(15))^x^(2-3)` is `3` b. `0` c. `2` d. none of these

Options

A. 3
B. 0
C. 2
D. none of these

Answer

To solve the equation \((31 + 8\sqrt{15})^{x^2 - 3} + 1 = (32 + 8\sqrt{15})^{x^2 - 3}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (31 + 8\sqrt{15})^{x^2 - 3} + 1 = (32 + 8\sqrt{15})^{x^2 - 3} \]

### Step 2: Identify the form of the equation
We can recognize that the left-hand side can be expressed using the identity \(a^n + b^n = (a + b)^n\) when \(n = 1\). We rewrite the equation as:
\[ (31 + 8\sqrt{15})^{x^2 - 3} + 1^{x^2 - 3} = (32 + 8\sqrt{15})^{x^2 - 3} \]

### Step 3: Set up the new equation
This implies:
\[ (31 + 8\sqrt{15})^{x^2 - 3} = (32 + 8\sqrt{15})^{x^2 - 3} - 1 \]

### Step 4: Apply the identity
We can apply the identity:
\[ a^{n} + b^{n} = (a + b)^{n} \quad \text{for } n = 1 \]
This gives us:
\[ (31 + 8\sqrt{15})^{x^2 - 3} + 1 = (32 + 8\sqrt{15})^{x^2 - 3} \]

### Step 5: Solve for \(x^2 - 3\)
From the equation, we can set:
\[ x^2 - 3 = 1 \]
This simplifies to:
\[ x^2 = 4 \]

### Step 6: Find the values of \(x\)
Taking the square root of both sides, we find:
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 7: Calculate the sum of values of \(x\)
Now, we calculate the sum of the values of \(x\):
\[ 2 + (-2) = 0 \]

### Conclusion
The sum of the values of \(x\) satisfying the equation is:
\[ \boxed{0} \]