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
Let's solve this problem again, Ferdinand. To calculate the value of (0.8)^4 using the binomial theorem, we first express 0.8 as a difference from 1. Step 1: Rewrite the expression in the form (a+b)^n. We can write 0.8 as 1 - 0.2. So, we need to expand (1 - 0.2)^4. Here, a=1, b=-0.2, and n=4. Step 2: Apply the binomial theorem formula. The binomial theorem states that (a+b)^n = _k=0^n nk a^n-k b^k. For (1 - 0.2)^4, the expansion is: (1 - 0.2)^4 = 40 (1)^4 (-0.2)^0 + 41 (1)^3 (-0.2)^1 + 42 (1)^2 (-0.2)^2 + 43 (1)^1 (-0.2)^3 + 44 (1)^0 (-0.2)^4 Step 3: Calculate the binomial coefficients. 40 = 1 41 = 4 42 = (4 × 3)/(2 × 1) = 6 43 = 4 44 = 1 Step 4: Substitute the coefficients and calculate each term. 1 · (1)^4 · (-0.2)^0 = 1 · 1 · 1 = 1 4 · (1)^3 · (-0.2)^1 = 4 · 1 · (-0.2) = -0.8 6 · (1)^2 · (-0.2)^2 = 6 · 1 · (0.04) = 0.24 4 · (1)^1 · (-0.2)^3 = 4 · 1 · (-0.008) = -0.032 1 · (1)^0 · (-0.2)^4 = 1 · 1 · (0.0016) = 0.0016 Step 5: Sum all the terms. (1 - 0.2)^4 = 1 - 0.8 + 0.24 - 0.032 + 0.0016 (1 - 0.2)^4 = 0.2 + 0.24 - 0.032 + 0.0016 (1 - 0.2)^4 = 0.44 - 0.032 + 0.0016 (1 - 0.2)^4 = 0.408 + 0.0016 (1 - 0.2)^4 = 0.4096 Last free one today — make it count tomorrow, or type /upgrade for unlimited.