For any integer n, the product (α^n)(β^n)(γ^n) equals which expression?

For any integer n, the product (α^n)(β^n)(γ^n) equals which expression?

• A. (αβ)^n
• B. (αβγ)^n
• C. (αβδ)^n
• D. (αβγδ)^n

Answer: (B)

Raising a product to a power distributes the exponent across the factors. Here, the same exponent n is applied to α, β, and γ, so their product raised to n is the nth power of αβγ. In symbols, α^n β^n γ^n = (αβγ)^n. This follows from (xy)^n = x^n y^n for integers n (and the fact that multiplication is associative and commutative here), giving α^n β^n γ^n = (αβ)^n γ^n = [(αβ) γ]^n = (αβγ)^n. Therefore, the correct expression is (αβγ)^n.

Raising a product to a power distributes the exponent across the factors. Here, the same exponent n is applied to α, β, and γ, so their product raised to n is the nth power of αβγ. In symbols, α^n β^n γ^n = (αβγ)^n. This follows from (xy)^n = x^n y^n for integers n (and the fact that multiplication is associative and commutative here), giving α^n β^n γ^n = (αβ)^n γ^n = [(αβ) γ]^n = (αβγ)^n. Therefore, the correct expression is (αβγ)^n.