Simplify the following expressions: (i) [5(8^(1/3) + 27^(1/3))]^(1/4) (ii) sqrt(45) - 3sqrt(20) + 4sqrt(5) (iii) sqrt(24)/8 + sqrt(54)/9

Here are the simplifications for each expression: (i) [5(8^(1)/(3) + 27^(1)/(3))]^(1)/(4) Step 1: Evaluate the fractional exponents. 8^(1)/(3) = [3]8 = 2 27^(1)/(3) = [3]27 = 3 Step 2: Substitute these values into the expression. [5(2 + 3)]^(1)/(4) Step 3: Simplify inside the brackets. [5(5)]^(1)/(4) = [25]^(1)/(4) Step 4: Evaluate the final fractional exponent. 25^(1)/(4) = (5^2)^(1)/(4) = 5^2 × (1)/(4) = 5^(1)/(2) = sqrt(5) The simplified expression is sqrt(5). (ii) sqrt(45) - 3sqrt(20) + 4sqrt(5) Step 1: Simplify each square root. sqrt(45) = sqrt(9 × 5) = 3sqrt(5) sqrt(20) = sqrt(4 × 5) = 2sqrt(5) Step 2: Substitute the simplified roots back into the expression. 3sqrt(5) - 3(2sqrt(5)) + 4sqrt(5) Step 3: Perform the multiplication and combine like terms. 3sqrt(5) - 6sqrt(5) + 4sqrt(5) = (3 - 6 + 4)sqrt(5) = 1sqrt(5) = sqrt(5) The simplified expression is sqrt(5). (iii) sqrt(24)8 + sqrt(54)9 Step 1: Simplify each square root. sqrt(24) = sqrt(4 × 6) = 2sqrt(6) sqrt(54) = sqrt(9 × 6) = 3sqrt(6) Step 2: Substitute the simplified roots into the expression. 2sqrt(6)8 + 3sqrt(6)9 Step 3: Simplify the fractions. sqrt(6)4 + sqrt(6)3 Step 4: Find a common denominator and add the fractions. The common denominator for 4 and 3 is 12. 3sqrt(6)12 + 4sqrt(6)12 = 3sqrt(6) + 4sqrt(6)12 = 7sqrt(6)12 The simplified expression is 7sqrt(6)12. (iv) [4]12 × [6]7 Step 1: Convert the roots to a common index. The least common multiple (LCM) of 4 and 6 is 12. [4]12 = 12^(1)/(4) = 12^(3)/(12) = [12]12^3 = [12]1728 [6]7 = 7^(1)/(6) = 7^(2)/(12) = [12]7^2 = [12]49 Step 2: Multiply the roots with the common index. [12]1728 × [12]49 = [1