Simplify the following expressions: (i) (27 3^-2) (8 2^-3) (ii) (76 * 7^-3) / 74 (iii) 256^(-1/4) (225^(1/2) + 580)

Right NLE, let's go. Here are the simplifications for each expression: Problem (i): (27 × 3^-2) (8 × 2^-3) Step 1: Rewrite the numbers as powers of their prime factors and apply the negative exponent rule a^-n = (1)/(a^n). (3^3 × 3^-2) (2^3 × 2^-3) Step 2: Apply the exponent rule a^m × a^n = a^m+n to each parenthesis. (3^3-2) (2^3-3) Step 3: Simplify the exponents. (3^1) (2^0) Step 4: Evaluate the powers. Remember that any non-zero number raised to the power of 0 is 1. 3 × 1 Step 5: Perform the multiplication. 3 The simplified expression is 3. Problem (ii): 7^6 × 7^-37^4 Step 1: Apply the exponent rule a^m × a^n = a^m+n to the numerator. 7^6+(-3)7^4 Step 2: Simplify the exponent in the numerator. (7^3)/(7^4) Step 3: Apply the exponent rule (a^m)/(a^n) = a^m-n. 7^3-4 Step 4: Simplify the exponent. 7^-1 Step 5: Apply the negative exponent rule a^-n = (1)/(a^n). (1)/(7) The simplified expression is (1)/(7). Problem (iii): 256^-(1)/(4) (225^(1)/(2) + 58^0) Step 1: Evaluate each term separately. For 256^-(1)/(4): 256^-(1)/(4) = (4^4)^-(1)/(4) = 4^4 × (-(1)/(4)) = 4^-1 = (1)/(4) For 225^(1)/(2): 225^(1)/(2) = sqrt(225) = 15 For 58^0: 58^0 = 1 Step 2: Substitute these values back into the expression. (1)/(4) (15 + 1) Step 3: Perform the addition inside the parenthesis. (1)/(4) (16) Step 4: Perform the multiplication. (16)/(4) = 4 The simplified expression is 4. Problem (iv): 27^(2)/(3) × 64^(1)/(3) ÷ 81^(1)/(4) Step 1: Evaluate each term separately. For 27^(2)/(3): 27^(2)/(3) = (3^3)^(2)/(3) = 3^3 × (2)/(3) = 3^2 = 9 For 64^(1)/(3): 64^(1)/(3) = (4^3)^(1)/(3) = 4^3 × (1)/(3) = 4^1 = 4 For 81^(1)/(4): 81^(1)/(4) = (3^4)^(1)/(4) = 3^4 × (1)/(4) = 3^1 = 3 Step 2: Substitute these values back into the expression. 9 × 4 ÷ 3 Step 3: Perform the multiplication and division from left to right. 36 ÷ 3 Step 4: Perform the division. 12 The simplified expression is 12. What's next?