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.
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Here's the solution for question 15. Question 15: Expand (4a^2 + 12a + 9)^3.5 up to fourth term, hence solve for (6.05)^7. Step 1: Simplify the expression inside the parenthesis. The expression 4a^2 + 12a + 9 is a perfect square trinomial. 4a^2 + 12a + 9 = (2a)^2 + 2(2a)(3) + 3^2 = (2a + 3)^2 Now substitute this back into the original expression: ( (2a + 3)^2 )^3.5 = (2a + 3)^2 × 3.5 = (2a + 3)^7 Step 2: Expand (2a + 3)^7 using the binomial theorem up to the fourth term. The binomial theorem states that (x + y)^n = _k=0^n nk x^n-k y^k. Here, x = 2a, y = 3, and n = 7. The first term (k=0): T_1 = 70 (2a)^7-0 (3)^0 = 1 · (2a)^7 · 1 = 128a^7 The second term (k=1): T_2 = 71 (2a)^7-1 (3)^1 = 7 · (2a)^6 · 3 = 7 · 64a^6 · 3 = 1344a^6 The third term (k=2): T_3 = 72 (2a)^7-2 (3)^2 = (7 × 6)/(2 × 1) · (2a)^5 · 9 = 21 · 32a^5 · 9 = 6048a^5 The fourth term (k=3): T_4 = 73 (2a)^7-3 (3)^3 = (7 × 6 × 5)/(3 × 2 × 1) · (2a)^4 · 27 = 35 · 16a^4 · 27 = 15120a^4 The expansion up to the fourth term is: 128a^7 + 1344a^6 + 6048a^5 + 15120a^4 + Step 3: Solve for (6.05)^7 using the binomial expansion method. To solve for (6.05)^7, we can write it as (6 + 0.05)^7. This is in the form (x+y)^n where x=6, y=0.05, and n=7. Since y is a small value, the first few terms of the expansion will give a good approximation. First term: T_1 = 70 (6)^7 (0.05)^0 = 1 · 279936 · 1 = 279936 Second term: T_2 = 71 (6)^6 (0.05)^1 = 7 · 46656 · 0.05 = 7 · 2332.8 = 16329.6 Third term: T_3 = 72 (6)^5 (0.05)^2 = 21 · 7776 · 0.0025 = 21 · 19.44 = 408.24 Fourth term: T_4 = 73 (6)^4 (0.05)^3 = 35 · 1296 · 0.000125 = 35 · 0.162 = 5.67 Step 4: Sum the first four terms to approximate (6.05)^7. (6.05)^7 ≈ 279936 + 16329.6 + 408.24 + 5.67 = 296679.51 The expansion up to the fourth term is 128a^7 + 1344a^6 + 6048a^5 + 15120a^4. The value of (6.05)^7 is approximately 296679.51. Expansion: 128a^7 + 1344a^6 + 6048a^5 + 15120a^4; Value of (6.05)^7 ≈ 296679.51 Last free one today — make it count tomorrow, or type /upgrade for unlimited.