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
To solve for x in the equation 9^(x-3) × 81^(1-x) = (1)/(27^x), we need to express all terms with the same base, which is 3. Step 1: Andika namba zote kama 'powers' za 3. Tunajua kuwa 9 = 3^2, 81 = 3^4, na 27 = 3^3. Hivyo, equation inakuwa: (3^2)^(x-3) × (3^4)^(1-x) = (1)/((3^3)^x) Step 2: Tumia kanuni ya 'power of a power', (a^m)^n = a^mn. 3^2(x-3) × 3^4(1-x) = (1)/(3^3x) 3^2x-6 × 3^4-4x = (1)/(3^3x) Step 3: Tumia kanuni ya kuzidisha 'powers' zenye 'base' sawa, a^m × a^n = a^m+n, kwa upande wa kushoto. 3^(2x-6) + (4-4x) = (1)/(3^3x) 3^-2x-2 = (1)/(3^3x) Step 4: Andika upande wa kulia kwa kutumia 'negative exponent', (1)/(a^n) = a^-n. 3^-2x-2 = 3^-3x Step 5: Kwa kuwa 'bases' ni sawa, tunalinganisha 'exponents'. -2x-2 = -3x Step 6: Tatua kwa x. -2x + 3x = 2 x = 2 Jibu la mwisho ni: x=2 Drop the next question!