![Simplify the following expressions involving square roots: a) (2\[[2]5+3)([2]5+2) b) ([2]6+2)([2]3+[2]2) c) (4[2]3+[2]6)(4[2]3-[2]6)](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1778068576854-3a5d665fdf60a95e.png&w=3840&q=75)
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 from Example 5: a) (2sqrt(5)+3)(sqrt(5)+2) Step 1: Expand the expression using the FOIL method. (2sqrt(5))(sqrt(5)) + (2sqrt(5))(2) + (3)(sqrt(5)) + (3)(2) 2(5) + 4sqrt(5) + 3sqrt(5) + 6 Step 2: Simplify and combine like terms. 10 + 7sqrt(5) + 6 16 + 7sqrt(5) The simplified expression is 16 + 7sqrt(5). b) (sqrt(6)+2)(sqrt(3)+sqrt(2)) Step 1: Expand the expression using the FOIL method. (sqrt(6))(sqrt(3)) + (sqrt(6))(sqrt(2)) + (2)(sqrt(3)) + (2)(sqrt(2)) sqrt(18) + sqrt(12) + 2sqrt(3) + 2sqrt(2) Step 2: Simplify the surds sqrt(18) and sqrt(12). sqrt(18) = sqrt(9 × 2) = 3sqrt(2) sqrt(12) = sqrt(4 × 3) = 2sqrt(3) Step 3: Substitute the simplified surds back into the expression and combine like terms. 3sqrt(2) + 2sqrt(3) + 2sqrt(3) + 2sqrt(2) (3sqrt(2) + 2sqrt(2)) + (2sqrt(3) + 2sqrt(3)) 5sqrt(2) + 4sqrt(3) The simplified expression is 5sqrt(2) + 4sqrt(3). c) (4sqrt(3)+sqrt(6))(4sqrt(3)-sqrt(6)) Step 1: This expression is in the form (a+b)(a-b), which simplifies to a^2 - b^2. Here, a = 4sqrt(3) and b = sqrt(6). (4sqrt(3))^2 - (sqrt(6))^2 Step 2: Calculate the squares. (4^2 × (sqrt(3))^2) - 6 (16 × 3) - 6 48 - 6 42 The simplified expression is 42. d) (3sqrt(3)-sqrt(7))^2 Step 1: Expand the expression using the formula (a-b)^2 = a^2 - 2ab + b^2. Here, a = 3sqrt(3) and b = sqrt(7). (3sqrt(3))^2 - 2(3sqrt(3))(sqrt(7)) + (sqrt(7))^2 Step 2: Calculate each term. (3^2 × (sqrt(3))^2) - (2 × 3 × sqrt(3 × 7)) + 7 (9 × 3) - 6sqrt(21) + 7 27 - 6sqrt(21) + 7 Step 3: Combine the constant terms. 34 - 6sqrt(21) The simplified expression is 34 - 6sqrt(21). e) (sqrt(3)+(1)/(sqrt(3)))^2 Step 1: Rationalize the term (1)/(sqrt(3)). (1)/(sqrt(3)) = (1)/(sqrt(3)) × sqrt(3)sqrt(3) = sqrt(3)3 Step 2: Substitute the rationalized term back into the expression. (sqrt(3)+sqrt(3)3)^2 Step 3: Combine the terms inside the parenthesis. (3sqrt(3)3+sqrt(3)3)^2 = (3sqrt(3)+sqrt(3)3)^2 = (4sqrt(3)3)^2 Step 4: Square the entire expression. (4sqrt(3))^23^2 = 4^2 × (sqrt(3))^29 (16 × 3)/(9) = (48)/(9) Step 5: Simplify the fraction. (48)/(9) = (16)/(3) The simplified expression is (16)/(3). That's 2 down. 3 left today — send the next one.