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
Peter_Schemer, let's knock this out. Here are the solutions for the exponential equations: 2. Solve for x: a) 27^x = 9^3x-2 Step 1: Express both bases as powers of 3. (3^3)^x = (3^2)^3x-2 Step 2: Apply the power rule (a^m)^n = a^mn. 3^3x = 3^2(3x-2) 3^3x = 3^6x-4 Step 3: Equate the exponents since the bases are the same. 3x = 6x-4 Step 4: Solve for x. 4 = 6x - 3x 4 = 3x x = (4)/(3) The solution is x = (4)/(3). b) 32^x+1 = 4^2x-1 Step 1: Express both bases as powers of 2. (2^5)^x+1 = (2^2)^2x-1 Step 2: Apply the power rule (a^m)^n = a^mn. 2^5(x+1) = 2^2(2x-1) 2^5x+5 = 2^4x-2 Step 3: Equate the exponents since the bases are the same. 5x+5 = 4x-2 Step 4: Solve for x. 5x - 4x = -2 - 5 x = -7 The solution is x = -7. c) 81^x-2 = 9^3x+1 Step 1: Express both bases as powers of 9. (9^2)^x-2 = 9^3x+1 Step 2: Apply the power rule (a^m)^n = a^mn. 9^2(x-2) = 9^3x+1 9^2x-4 = 9^3x+1 Step 3: Equate the exponents since the bases are the same. 2x-4 = 3x+1 Step 4: Solve for x. -4 - 1 = 3x - 2x -5 = x The solution is x = -5. 3. Solve for x: a) 3^2x+1 - 9^x = 162 Step 1: Rewrite terms with a common base. Note that 9 = 3^2. 3^2x · 3^1 - (3^2)^x = 162 3 · (3^x)^2 - (3^x)^2 = 162 Step 2: Let y = 3^x. Substitute y into the equation. 3y^2 - y^2 = 162 2y^2 = 162 Step 3: Solve for y. y^2 = (162)/(2) y^2 = 81 y = ±sqrt(81) y = ± 9 Step 4: Substitute back y = 3^x. Since 3^x must be positive, we take y=9. 3^x = 9 3^x = 3^2 x = 2 The solution is x = 2. b) 8^x+1 - 2^3x-1 = 120 Step 1: Rewrite terms with a common base. Note that 8 = 2^3. (2^3)^x+1 - 2^3x-1 = 120 2^3(x+1) - 2^3x-1 = 120 2^3x+3 - 2^3x-1 = 120 Step 2: Factor out 2^3x using the rule a^m+n = a^m · a^n. 2^3x · 2^3 - 2^3x · 2^-1 = 120 2^3x (2^3 - 2^-1) = 120 2^3x ( 8 - (1)/(2) ) = 120 Step 3: Simplify the expression in the parenthesis. 2^3x ( (16)/(2) - (1)/(2) ) = 120 2^3x ( (15)/(2) ) = 120 Step 4: Solve for 2^3x. 2^3x = 120 · (2)/(15) 2^3x = (240)/(15) 2^3x = 16 Step 5: Express 16 as a power of 2 and solve for x. 2^3x = 2^4 3x = 4 x = (4)/(3) The solution is x = (4)/(3). c) 7^2x+1 - 49^x = 294 Step 1: Rewrite terms with a common base. Note that 49 = 7^2. 7^2x · 7^1 - (7^2)^x = 294 7 · (7^x)^2 - (7^x)^2 = 294 Step 2: Let y = 7^x. Substitute y into the equation. 7y^2 - y^2 = 294 6y^2 = 294 Step 3: Solve for y. y^2 = (294)/(6) y^2 = 49 y = ±sqrt(49) y = ± 7 Step 4: Substitute back y = 7^x. Since 7^x must be positive, we take y=7. 7^x = 7 7^x = 7^1 x = 1 The solution is x = 1. Send me the next one 📸