What is the value of the following? Assume x > 0 and a, b, and c are real…
2023
What is the value of the following?
Assume x > 0 and a, b, and c are real numbers.
xa(b − c) · xb(c − a) · xc(a − b)
Answer: B. 1 — ConceptWhen powers with the same nonzero base are multiplied, their exponents are added: xm · xn = xm + n. If the combined exponent is zero, the zero-exponent…
- A.
0
- B.
1
- C.
x
- D.
More than one of the above
- E.
None of the above
Attempted by 7 students.
Show answer & explanation
Correct answer: B
Concept
When powers with the same nonzero base are multiplied, their exponents are added: xm · xn = xm + n. If the combined exponent is zero, the zero-exponent rule gives x0 = 1.
Application
Combine the exponents because the base x is common: a(b − c) + b(c − a) + c(a − b).
Expand the brackets: ab − ac + bc − ab + ac − bc.
Cancel equal opposite terms: ab − ab = 0, −ac + ac = 0, and bc − bc = 0. Thus the combined exponent is 0.
Cross-check
Substitute a = 1, b = 2, and c = 3. The exponents are −1, 4, and −3, whose sum is 0; the product again becomes x0.
Result
Therefore, for x > 0, the expression simplifies to 1.