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

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

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

Answer: (D)

Raising a product to the same power distributes over multiplication: (xy)^n = x^n y^n, and multiplication is associative. Apply this step by step to (α^n)(β^n)(γ^n)(δ^n). Combine the first two: (α^n)(β^n) = (αβ)^n. Then multiply by γ^n: (αβ)^n(γ^n) = (αβγ)^n. Finally multiply by δ^n: (αβγ)^n(δ^n) = (αβγδ)^n. So the product equals (αβγδ)^n. The other forms would miss one or more bases inside the nth power, so they don’t match the original product.

Raising a product to the same power distributes over multiplication: (xy)^n = x^n y^n, and multiplication is associative.

Apply this step by step to (α^n)(β^n)(γ^n)(δ^n). Combine the first two: (α^n)(β^n) = (αβ)^n. Then multiply by γ^n: (αβ)^n(γ^n) = (αβγ)^n. Finally multiply by δ^n: (αβγ)^n(δ^n) = (αβγδ)^n. So the product equals (αβγδ)^n. The other forms would miss one or more bases inside the nth power, so they don’t match the original product.