What value of n makes the equation true? (2 · xⁿ · yⁿ)(4 · x² · y¹⁰) = 8 · x¹¹ · y²⁰
Answer
By simplifying the equation and equating the exponents on both sides, we find that the value of n that makes the equation true is n=9.
Explanation
To solve for the value of n in the equation (2 · xⁿ · yⁿ)(4 · x² · y¹⁰) = 8 · x¹¹ · y²⁰, we use the product rule for exponents, which states that aᵐ · aⁿ = aᵐ⁺ⁿ. 1. Simplify the left-hand side: Multiply the coefficients and the variables separately: (2 · 4) · (xⁿ · x²) · (yⁿ · y¹⁰) = 8 · xⁿ⁺² · yⁿ⁺¹⁰ 2. Set the simplified left side equal to the right side: 8 · xⁿ⁺² · yⁿ⁺¹⁰ = 8 · x¹¹ · y²⁰ 3. Equate the exponents: Since the coefficients (8) and bases (x, y) are the same on both sides, the exponents must be equal. For x: n + 2 = 11 n = 11 - 2 = 9 For y: n + 10 = 20 n = 10 In conclusion, the value of n that satisfies the equation is 9. While y is also determined to be 10, the specific requirement of the question is to solve for n.