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
Step 1: Let x = [3]2 . Note that [3]4 = x^2 , since 4 = 2^2 . [3]2 + [3]43[3]2 + 2[3]4 = (x + x^2)/(3x + 2x^2) Step 2: Factor x from the numerator and denominator. Numerator: x + x^2 = x(1 + x) Denominator: 3x + 2x^2 = x(3 + 2x) (x(1 + x))/(x(3 + 2x)) = (1 + x)/(3 + 2x) Step 3: Rationalize the denominator by multiplying the numerator and denominator by 9 - 6x + 4x^2 , using the identity (3 + 2x)(9 - 6x + 4x^2) = 43. (1 + x)/(3 + 2x) = ((1 + x)(9 - 6x + 4x^2))/(43) Step 4: Expand the numerator. (1 + x)(9 - 6x + 4x^2) = 1 · 9 + 1 · (-6x) + 1 · 4x^2 + x · 9 + x · (-6x) + x · 4x^2 = 9 - 6x + 4x^2 + 9x - 6x^2 + 4x^3 Step 5: Substitute x^3 = 2 , so 4x^3 = 8 . Combine like terms: 9 + 8 + (-6x + 9x) + (4x^2 - 6x^2) = 17 + 3x - 2x^2 Step 6: Substitute back x = [3]2 and x^2 = [3]4 . 17 + 3[3]2 - 2[3]443 17 + 3[3]2 - 2[3]443