g) Rationalize (2)/(3sqrt(5)+4)

Here are the solutions to the surd rationalization problems: g) Rationalize (2)/(3sqrt(5)+4) Step 1: Identify the conjugate of the denominator. The denominator is 3sqrt(5)+4. Its conjugate is 3sqrt(5)-4. Step 2: Multiply the numerator and denominator by the conjugate. (2)/(3sqrt(5)+4) × 3sqrt(5)-43sqrt(5)-4 Step 3: Simplify the numerator. 2(3sqrt(5)-4) = 6sqrt(5) - 8 Step 4: Simplify the denominator using the difference of squares formula (a+b)(a-b) = a^2 - b^2. Here, a = 3sqrt(5) and b = 4. (3sqrt(5))^2 - (4)^2 = (3^2 × (sqrt(5))^2) - 16 = (9 × 5) - 16 = 45 - 16 = 29 Step 5: Write the rationalized expression. 6sqrt(5)-829 The rationalized form is 6sqrt(5)-829. h) Rationalize (6)/(2sqrt(2)-1) Step 1: Identify the conjugate of the denominator. The denominator is 2sqrt(2)-1. Its conjugate is 2sqrt(2)+1. Step 2: Multiply the numerator and denominator by the conjugate. (6)/(2sqrt(2)-1) × 2sqrt(2)+12sqrt(2)+1 Step 3: Simplify the numerator. 6(2sqrt(2)+1) = 12sqrt(2) + 6 Step 4: Simplify the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. Here, a = 2sqrt(2) and b = 1. (2sqrt(2))^2 - (1)^2 = (2^2 × (sqrt(2))^2) - 1 = (4 × 2) - 1 = 8 - 1 = 7 Step 5: Write the rationalized expression. 12sqrt(2)+67 The rationalized form is 12sqrt(2)+67. i) Rationalize 2sqrt(3)+2sqrt(3)-2 Step 1: Identify the conjugate of the denominator. The denominator is sqrt(3)-2. Its conjugate is sqrt(3)+2. Step 2: Multiply the numerator and denominator by the conjugate. 2sqrt(3)+2sqrt(3)-2 × sqrt(3)+2sqrt(3)+2 Step 3: Simplify the numerator using FOIL (First, Outer, Inner, Last). (2sqrt(3)+2)(sqrt(3)+2) = (2sqrt(3))(sqrt(3)) + (2sqrt(3))(2) + (2)(sqrt(3)) + (2)(2) = (2 × 3) + 4sqrt(3) + 2sqrt(3) + 4 = 6 + 6sqrt(3) + 4 = 10 + 6sqrt(3) Step 4: Simplify the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. Here, a = sqrt(3) and b = 2. (sqrt(3))^2 - (2)^2 = 3 - 4 = -1 Step 5: Write the rationalized expression. 10+6sqrt(3)-1 = -(10+6sqrt(3)) = -10-6sqrt(3) The rationalized form is -10-6sqrt(3). j) Rationalize 2+sqrt(5)2sqrt(5)-1 Step 1: Identify the conjugate of the denominator. The denominator is 2sqrt(5)-1. Its conjugate is 2sqrt(5)+1. Step 2: Multiply the numerator and denominator by the conjugate. 2+sqrt(5)2sqrt(5)-1 × 2sqrt(5)+12sqrt(5)+1 Step 3: Simplify the numerator using FOIL (First, Outer, Inner, Last). (2+sqrt(5))(2sqrt(5)+1) = (2)(2sqrt(5)) + (2)(1) + (sqrt(5))(2sqrt(5)) + (sqrt(5))(1) = 4sqrt(5) + 2 + (2 × 5) + sqrt(5) = 4sqrt(5) + 2 + 10 + sqrt(5) = 12 + 5sqrt(5) Step 4: Simplify the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. Here, a = 2sqrt(5) and b = 1. (2sqrt(5))^2 - (1)^2 = (2^2 × (sqrt(5))^2) - 1 = (4 × 5) - 1 = 20 - 1 = 19 Step 5: Write the rationalized expression. 12+5sqrt(5)19 The rationalized form is 12+5sqrt(5)19.