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
Problem 8 Step 1: Recall the difference of cubes formula n^3 - m^3 = (n - m)(n^2 + nm + m^2). Thus, (n^3 - m^3)/(n - m) = n^2 + nm + m^2. Step 2: Substitute n = 4 and m = 1: n^2 = 4^2 = 16, nm = 4 · 1 = 4, m^2 = 1^2 = 1. Step 3: Add the terms: 16 + 4 + 1 = 21. 21 Problem 9 Step 1: Compute r - s: r - s = 3 - 1 = 2. Step 2: Compute t - s: t - s = 2 - 1 = 1. Step 3: Compute the numerator (r - s)^3 - (t - s)^2: (r - s)^3 = 2^3 = 8, (t - s)^2 = 1^2 = 1, 8 - 1 = 7. Step 4: Divide by the denominator r - s: (7)/(2). (7)/(2) Problem 10 Step 1: Factor the expression algebraically: (ab)^3 + (bc)^3 = a^3 b^3 + b^3 c^3 = b^3 (a^3 + c^3), ((ab)^3 + (bc)^3)/(abc) = (b^3 (a^3 + c^3))/(abc) = (b^2 (a^3 + c^3))/(ac). Step 2: Substitute a=5, b=2, c=3: a^3 = 5^3 = 125, c^3 = 3^3 = 27, a^3 + c^3 = 125 + 27 = 152, b^2 = 2^2 = 4, ac = 5 · 3 = 15. Step 3: Multiply and divide: 4 · 152 = 608, (608)/(15). (608)/(15)