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
Here are the solutions to the problems. a) Evaluate 9^-1 × ((3)/(4))^2 Step 1: Evaluate 9^-1. 9^-1 = (1)/(9) Step 2: Evaluate ((3)/(4))^2. ((3)/(4))^2 = (3^2)/(4^2) = (9)/(16) Step 3: Multiply the results. (1)/(9) × (9)/(16) = (1 × 9)/(9 × 16) = (9)/(144) = (1)/(16) The evaluated expression is (1)/(16). b) Using indices find the value of x a) 16^x = 8^x+1 Step 1: Express both sides with base 2. 16 = 2^4 8 = 2^3 So the equation becomes: (2^4)^x = (2^3)^x+1 Step 2: Apply the power rule (a^m)^n = a^mn. 2^4x = 2^3(x+1) 2^4x = 2^3x+3 Step 3: Equate the exponents since the bases are the same. 4x = 3x+3 Step 4: Solve for x. 4x - 3x = 3 x = 3 The value of x is 3. b) 81^3x × 27^2x = 729 Step 1: Express all numbers with base 3. 81 = 3^4 27 = 3^3 729 = 3^6 So the equation becomes: (3^4)^3x × (3^3)^2x = 3^6 Step 2: Apply the power rule (a^m)^n = a^mn. 3^4 × 3x × 3^3 × 2x = 3^6 3^12x × 3^6x = 3^6 Step 3: Apply the product rule a^m × a^n = a^m+n. 3^12x+6x = 3^6 3^18x = 3^6 Step 4: Equate the exponents since the bases are the same. 18x = 6 Step 5: Solve for x. x = (6)/(18) x = (1)/(3) The value of x is (1)/(3). c) (64 × 128)/(8) = 2^2x-4 Step 1: Express all numbers with base 2. 64 = 2^6 128 = 2^7 8 = 2^3 So the equation becomes: (2^6 × 2^7)/(2^3) = 2^2x-4 Step 2: Simplify the left side using index rules. First, apply the product rule a^m × a^n = a^m+n in the numerator: 2^6+72^3 = 2^2x-4 2^132^3 = 2^2x-4 Next, apply the quotient rule (a^m)/(a^n) = a^m-n: 2^13-3 = 2^2x-4 2^10 = 2^2x-4 Step 3: Equate the exponents since the bases are the same. 10 = 2x-4 Step 4: Solve for x. 10 + 4 = 2x 14 = 2x x = (14)/(2) x = 7 The value of x is 7. Send me the next one 📸