Answer :
Answer:
Let's simplify the given expression step by step:
\[ \frac{x^b}{x^c}^{(b+c-a)} \times \frac{x^c}{x^a}^{(c+a-b)} \times \frac{x^a}{x^b}^{(a+b-c)} \]
First, simplify each term separately:
1. \( \frac{x^b}{x^c}^{(b+c-a)} = x^{b - c} \cdot (b + c - a) \)
2. \( \frac{x^c}{x^a}^{(c+a-b)} = x^{c - a} \cdot (c + a - b) \)
3. \( \frac{x^a}{x^b}^{(a+b-c)} = x^{a - b} \cdot (a + b - c) \)
Now, multiply these simplified expressions together:
\[ x^{(b - c) \cdot (b + c - a) + (c - a) \cdot (c + a - b) + (a - b) \cdot (a + b - c)} \]
Next, simplify the exponents:
Expand each term:
- \( (b - c) \cdot (b + c - a) = b^2 - c^2 - ab + ac \)
- \( (c - a) \cdot (c + a - b) = c^2 - a^2 - bc + ba \)
- \( (a - b) \cdot (a + b - c) = a^2 - b^2 - ac + bc \)
Add them together:
\[ b^2 - c^2 - ab + ac + c^2 - a^2 - bc + ba + a^2 - b^2 - ac + bc \]
Simplify by canceling out terms:
\[ - ab + ac + ba - bc \]
Combine like terms:
\[ a(b - c) + b(c - a) + c(a - b) \]
Therefore, the simplified expression is:
\[ x^{a(b - c) + b(c - a) + c(a - b)} \]
This is the simplified form of the given expression involving exponents and rational numbers.