Rationalize the denominator of the following:

Here are the solutions for rationalizing the denominator for each expression: 1. Rationalize the denominator of the following: i) (1)/(2+sqrt(3)) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 2-sqrt(3). (1)/(2+sqrt(3)) × 2-sqrt(3)2-sqrt(3) Step 2: Apply the difference of squares formula (a+b)(a-b) = a^2-b^2 in the denominator. 2-sqrt(3)(2)^2 - (sqrt(3))^2 = 2-sqrt(3)4-3 Step 3: Simplify the expression. 2-sqrt(3)1 = 2-sqrt(3) The rationalized expression is 2-sqrt(3). ii) (1)/(3+2sqrt(2)) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 3-2sqrt(2). (1)/(3+2sqrt(2)) × 3-2sqrt(2)3-2sqrt(2) Step 2: Apply the difference of squares formula in the denominator. 3-2sqrt(2)(3)^2 - (2sqrt(2))^2 = 3-2sqrt(2)9 - (4 × 2) Step 3: Simplify the expression. 3-2sqrt(2)9-8 = 3-2sqrt(2)1 = 3-2sqrt(2) The rationalized expression is 3-2sqrt(2). iii) (1)/(4sqrt(3)-5) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 4sqrt(3)+5. (1)/(4sqrt(3)-5) × 4sqrt(3)+54sqrt(3)+5 Step 2: Apply the difference of squares formula in the denominator. 4sqrt(3)+5(4sqrt(3))^2 - (5)^2 = 4sqrt(3)+5(16 × 3) - 25 Step 3: Simplify the expression. 4sqrt(3)+548-25 = 4sqrt(3)+523 The rationalized expression is 4sqrt(3)+523. iv) (16)/(2sqrt(3)+11) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 2sqrt(3)-sqrt(11). (16)/(2sqrt(3)+11) × 2sqrt(3)-sqrt(11)2sqrt(3)-sqrt(11) Step 2: Apply the difference of squares formula in the denominator. 16(2sqrt(3)-sqrt(11))(2sqrt(3))^2 - (sqrt(11))^2 = 16(2sqrt(3)-sqrt(11))(4 × 3) - 11 Step 3: Simplify the expression. 16(2sqrt(3)-sqrt(11))12-11 = 16(2sqrt(3)-sqrt(11))1 = 16(2sqrt(3)-sqrt(11)) = 32sqrt(3)-16sqrt(11) The rationalized expression is 32sqrt(3)-16sqrt(11). v) 9-sqrt(2)9+sqrt(2) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 9-sqrt(2). 9-sqrt(2)9+sqrt(2) × 9-sqrt(2)9-sqrt(2) Step 2: Expand the numerator using (a-b)^2 = a^2-2ab+b^2 and the denominator using (a+b)(a-b) = a^2-b^2. (9-sqrt(2))^2(9)^2 - (sqrt(2))^2 = 9^2 - 2(9)(sqrt(2)) + (sqrt(2))^281 - 2 Step 3: Simplify the expression. 81 - 18sqrt(2) + 279 = 83 - 18sqrt(2)79 The rationalized expression is 83 - 18sqrt(2)79. vi) sqrt(13)+sqrt(11)sqrt(13)-sqrt(11) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(13)+sqrt(11). sqrt(13)+sqrt(11)sqrt(13)-sqrt(11) × sqrt(13)+sqrt(11)sqrt(13)+sqrt(11) Step 2: Expand the numerator using (a+b)^2 = a^2+2ab+b^2 and the denominator using (a-b)(a+b) = a^2-b^2. (sqrt(13)+sqrt(11))^2(sqrt(13))^2 - (sqrt(11))^2 = (sqrt(13))^2 + 2(sqrt(13))(sqrt(11)) + (sqrt(11))^213 - 11 Step 3: Simplify the expression. 13 + 2sqrt(143) + 112 = 24 + 2sqrt(143)2 Step 4: Factor out 2 from the numerator and simplify. 2(12 + sqrt(143))2 = 12 + sqrt(143) The rationalized expression is 12 + sqrt(143). Last free one today — make it count tomorrow, or type /upgrade for unlimited.