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: Express all numbers in the equation as powers of 3. The given equation is: 27^x × 9^2x((1)/(3))^x = 81 We can rewrite the bases as powers of 3: 27 = 3^3 9 = 3^2 (1)/(3) = 3^-1 81 = 3^4 Substitute these into the equation: (3^3)^x × (3^2)^2x(3^-1)^x = 3^4 Step 2: Simplify the exponents using the rule (a^m)^n = a^mn. 3^3x × 3^2 × 2x3^-x = 3^4 3^3x × 3^4x3^-x = 3^4 Step 3: Combine the terms in the numerator using the rule a^m × a^n = a^m+n. 3^3x + 4x3^-x = 3^4 3^7x3^-x = 3^4 Step 4: Simplify the fraction using the rule (a^m)/(a^n) = a^m-n. 3^7x - (-x) = 3^4 3^7x + x = 3^4 3^8x = 3^4 Step 5: Equate the exponents since the bases are the same. 8x = 4 Step 6: Solve for x. x = (4)/(8) x = (1)/(2) The value of x is (1)/(2) 3 done, 2 left today. You're making progress.