Simplify sqrt(8).

Fresh day More, let's solve. a) To simplify the expression, we first simplify sqrt(8) and then rationalize the denominator. Step 1: Simplify sqrt(8). sqrt(8) = sqrt(4 × 2) = 2sqrt(2) Step 2: Substitute the simplified term into the expression. 2sqrt(2)-24-sqrt(2) Step 3: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 4+sqrt(2). 2sqrt(2)-24-sqrt(2) × 4+sqrt(2)4+sqrt(2) Step 4: Expand the numerator. (2sqrt(2)-2)(4+sqrt(2)) = 2sqrt(2)(4) + 2sqrt(2)(sqrt(2)) - 2(4) - 2(sqrt(2)) = 8sqrt(2) + 2(2) - 8 - 2sqrt(2) = 8sqrt(2) + 4 - 8 - 2sqrt(2) = (8sqrt(2) - 2sqrt(2)) + (4 - 8) = 6sqrt(2) - 4 Step 5: Expand the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2. (4-sqrt(2))(4+sqrt(2)) = 4^2 - (sqrt(2))^2 = 16 - 2 = 14 Step 6: Combine the simplified numerator and denominator. 6sqrt(2)-414 Step 7: Simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 2. 2(3sqrt(2)-2)2(7) = 3sqrt(2)-27 The simplified expression is 3sqrt(2)-27. b) To express the given expression as a single fraction, we first factor the denominator of the first term. Step 1: Factor the denominator y^2-y-2. We look for two numbers that multiply to -2 and add to -1. These numbers are -2 and 1. y^2-y-2 = (y-2)(y+1) Step 2: Rewrite the expression with the factored denominator. (1)/((y-2)(y+1)) - (1)/(y+1) Step 3: Find a common denominator, which is (y-2)(y+1). To get this common denominator for the second term, multiply its numerator and denominator by (y-2). (1)/(y+1) = (1)/(y+1) × (y-2)/(y-2) = (y-2)/((y-2)(y+1)) Step 4: Perform the subtraction. (1)/((y-2)(y+1)) - (y-2)/((y-2)(y+1)) = (1 - (y-2))/((y-2)(y+1)) Step 5: Simplify the numerator. 1 - (y-2) = 1 - y + 2 = 3 - y Step 6: Write the final single fraction. (3-y)/((y-2)(y+1)) The expression as a single fraction is (3-y)/((y-2)(y+1)). a) Let the first term of the Arithmetic Progression (AP) be a and the common difference be d. The first three terms are a, a+d, and a+2d. Step 1: Use the information about the average of the first three terms. The average of the first three terms is 56. (a + (a+d) + (a+2d))/(3) = 56 (3a + 3d)/(3) = 56 a + d = 56 (*) Step 2: Use the information about the sum of the first two terms. The sum of the first two terms is 96. a + (a+d) = 96 2a + d = 96 (**) Step 3: Solve the system of linear equations for a and d. Subtract equation () from equation (*): (2a + d) - (a + d) = 96 - 56 a = 40 Step 4: Substitute the value of a into equation (*) to find d. 40 + d = 56 d = 56 - 40 d = 16 The first term is 40 and the common difference is 16. Got more? Send 'em.