Simplify (x^b)^{(b + c − a)} · (x^c)^{(c + a − b)} · (x^a)^{(a + b − c)} to a single power of x.

Given data

• Expression: (x^b)^{(b + c − a)} · (x^c)^{(c + a − b)} · (x^a)^{(a + b − c)}

Concept / Approach

• Use (x^p)^q = x^{pq}. When multiplying same bases, add exponents.

Step-by-step exponent aggregation

Exponent of x = b(b + c − a) + c(c + a − b) + a(a + b − c)= b^2 + bc − ab + c^2 + ca − bc + a^2 + ab − acCross terms cancel: (bc − bc), (−ab + ab), (ca − ac) ⇒ remaining = a^2 + b^2 + c^2.

Verification

Pick a = 1, b = 2, c = 3: exponent sum = 1 + 4 + 9 = 14; numeric expansion confirms.

Common pitfalls

• Forgetting to distribute negatives in (b + c − a), etc.

Final Answer

Simplified form: x^{a^2 + b^2 + c^2}.