This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back MT — been a while! Let's pick up where you left off. Step 1: Expand the given expression (1 + (2)/(3)x)^8 using the binomial theorem. The binomial theorem states that (1+y)^n = 1 + ny + (n(n-1))/(2!)y^2 + . Here, n=8 and y=(2)/(3)x. (1 + (2)/(3)x)^8 = 1 + 8((2)/(3)x) + (8(8-1))/(2!)((2)/(3)x)^2 + = 1 + (16)/(3)x + (8 × 7)/(2)((4)/(9)x^2) + = 1 + (16)/(3)x + 28((4)/(9)x^2) + = 1 + (16)/(3)x + (112)/(9)x^2 + Step 2: Compare the expanded form with the given approximation 1 + (4)/(3)ax + (2)/(3)bx^2 + . By comparing the coefficients of x: (4)/(3)a = (16)/(3) Multiply both sides by 3: 4a = 16 a = (16)/(4) a = 4 Step 3: Compare the coefficients of x^2: (2)/(3)b = (112)/(9) Multiply both sides by 3: 2b = (112)/(9) × 3 2b = (112)/(3) Divide both sides by 2: b = (112)/(3 × 2) b = (112)/(6) Simplify the fraction: b = (56)/(3) The values of the constants are a=4 and b=(56)/(3). The final answer is a=4, b=(56)/(3). Send me the next one 📸