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.
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Huu hapa ni utatuzi wa swali la 1, sehemu a, b, c, na d: 1. Tatua viashiria vifuatavyo ukiacha jibu lako katika mfumo wa viashiria: a) (243 × 49)/(7) Step 1: Andika nambari zote kama nguvu za nambari kuu. 243 = 3 × 3 × 3 × 3 × 3 = 3^5 49 = 7 × 7 = 7^2 7 = 7^1 Step 2: Badilisha nambari hizi kwenye usemi. (3^5 × 7^2)/(7^1) Step 3: Tumia sheria ya viashiria (a^m)/(a^n) = a^m-n. 3^5 × 7^2-1 = 3^5 × 7^1 Jibu katika mfumo wa viashiria ni: 3^5 × 7^1 b) (1024 × 256)/(64 × 4) Step 1: Andika nambari zote kama nguvu za 2. 1024 = 2^10 256 = 2^8 64 = 2^6 4 = 2^2 Step 2: Badilisha nambari hizi kwenye usemi. 2^10 × 2^82^6 × 2^2 Step 3: Tumia sheria ya viashiria a^m × a^n = a^m+n kwa namba ya juu na namba ya chini. Namba ya juu: 2^10 × 2^8 = 2^10+8 = 2^18 Namba ya chini: 2^6 × 2^2 = 2^6+2 = 2^8 Usemi unakuwa: 2^182^8 Step 4: Tumia sheria ya viashiria (a^m)/(a^n) = a^m-n. 2^18-8 = 2^10 Jibu katika mfumo wa viashiria ni: 2^10 c) ((27)/(8))^x+7 = ((9)/(4))^30x Step 1: Andika besi kama nguvu za besi ya kawaida. (27)/(8) = (3^3)/(2^3) = ((3)/(2))^3 (9)/(4) = (3^2)/(2^2) = ((3)/(2))^2 Step 2: Badilisha hizi kwenye mlinganyo. (((3)/(2))^3)^x+7 = (((3)/(2))^2)^30x Step 3: Tumia sheria ya viashiria (a^m)^n = a^mn. ((3)/(2))^3(x+7) = ((3)/(2))^2(30x) ((3)/(2))^3x+21 = ((3)/(2))^60x Step 4: Kwa kuwa besi ni sawa, viashiria lazima viwe sawa. 3x + 21 = 60x Step 5: Tatua kwa x. 21 = 60x - 3x 21 = 57x x = (21)/(57) Step 6: Rahisisha sehemu. x = (21 ÷ 3)/(57 ÷ 3) = (7)/(19) Jibu ni: x = (7)/(19) d) (625)^-x ÷ (3125)^2x = 15625^x Step 1: Andika nambari zote kama nguvu za 5. 625 = 5^4 3125 = 5^5 15625 = 5^6 Step 2: Badilisha hizi kwenye mlinganyo. (5^4)^-x ÷ (5^5)^2x = (5^6)^x Step 3: Tumia sheria ya viashiria (a^m)^n = a^mn. 5^-4x ÷ 5^10x = 5^6x Step 4: Tumia sheria ya viashiria (a^m)/(a^n) = a^m-n kwa upande wa kushoto. 5^-4x - 10x = 5^6x 5^-14x = 5^6x Step 5: Kwa kuwa besi ni sawa, viashiria lazima viwe sawa. -14x = 6x Step 6: Tatua kwa x. 0 = 6x + 14x 0 = 20x x = (0)/(20) x = 0 Jibu ni: x = 0 That's 2 down. 3 left today — send the next one.