The value of `( x ^(b + c))^(b - c) ( x ^(c + a))^(c - a) ( x^( a + b))^(a - b) (x ne 0)` is

Options

A. A)1
B. B)2
C. C)`-1`
D. D)0

Answer

To solve the expression \(( x ^{(b + c)})^{(b - c)} ( x ^{(c + a)})^{(c - a)} ( x^{(a + b)})^{(a - b)}\) where \(x \neq 0\), we will follow these steps:

### Step 1: Apply the Power of a Power Rule
Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify each term in the expression.

1. \((x^{(b+c)})^{(b-c)} = x^{(b+c)(b-c)}\)
2. \((x^{(c+a)})^{(c-a)} = x^{(c+a)(c-a)}\)
3. \((x^{(a+b)})^{(a-b)} = x^{(a+b)(a-b)}\)

### Step 2: Rewrite the Expression
Now we can rewrite the entire expression as:
\[ x^{(b+c)(b-c)} \cdot x^{(c+a)(c-a)} \cdot x^{(a+b)(a-b)} \]

### Step 3: Combine the Exponents
Since the bases are the same (all are \(x\)), we can combine the exponents:
\[ x^{(b+c)(b-c) + (c+a)(c-a) + (a+b)(a-b)} \]

### Step 4: Expand Each Term
Now we will expand each of the terms in the exponent:
1. \((b+c)(b-c) = b^2 - c^2\)
2. \((c+a)(c-a) = c^2 - a^2\)
3. \((a+b)(a-b) = a^2 - b^2\)

### Step 5: Substitute Back into the Expression
Substituting these expansions back, we have:
\[ x^{(b^2 - c^2) + (c^2 - a^2) + (a^2 - b^2)} \]

### Step 6: Simplify the Exponent
Now we simplify the exponent:
\[ (b^2 - c^2) + (c^2 - a^2) + (a^2 - b^2) \]
Notice that:
- The \( -c^2 \) and \( +c^2 \) cancel out.
- The \( -a^2 \) and \( +a^2 \) cancel out.
- The \( -b^2 \) and \( +b^2 \) cancel out.

Thus, we are left with:
\[ 0 \]

### Step 7: Final Expression
So, we have:
\[ x^0 \]

### Conclusion
Since any non-zero number raised to the power of 0 is 1, we conclude:
\[ \text{The value is } 1. \]