Simplify the expression: \(2 - (y5)/(4)y(1)/(4) - y^-(1)/(4)\).

Here are the solutions to the problems. Question 5: Simplify (2 - (y^5)/(4)y^(1)/(4) - y^-(1)/(4)). Step 1: Simplify the denominator of the fraction. y^(1)/(4) - y^-(1)/(4) = y^(1)/(4) - (1)/(y^1)4 Find a common denominator: = ((y^1)/(4))^2 - 1y^(1)/(4) = (y^1)/(2) - 1y^(1)/(4) Step 2: Substitute the simplified denominator back into the fraction and simplify. (y^5)/(4)y^(1)/(4) - y^-(1)/(4) = (y^5)/(4)(y^1)/(2) - 1y^(1)/(4) Multiply by the reciprocal of the denominator: = y^(5)/(4) × (y^1)/(4)y^(1)/(2) - 1 Combine the terms in the numerator using exponent rules (a^m × a^n = a^m+n): = (y^5)/(4) + (1)/(4)y^(1)/(2) - 1 = (y^6)/(4)y^(1)/(2) - 1 = (y^3)/(2)y^(1)/(2) - 1 Step 3: Substitute this simplified fraction back into the original expression and combine terms. 2 - (y^3)/(2)y^(1)/(2) - 1 Find a common denominator, which is y^(1)/(2) - 1: = (2(y^1)/(2) - 1)y^(1)/(2) - 1 - (y^3)/(2)y^(1)/(2) - 1 = (2y^1)/(2) - 2 - y^(3)/(2)y^(1)/(2) - 1 Rearrange the terms in the numerator: = (-y^3)/(2) + 2y^(1)/(2) - 2y^(1)/(2) - 1 This can also be written by multiplying the numerator and denominator by -1: = (y^3)/(2) - 2y^(1)/(2) + 21 - y^(1)/(2) The simplified expression is (y^3)/(2) - 2y^(1)/(2) + 21 - y^(1)/(2). Question 6: Split (1)/((x-1)(x+2)) into partial fractions. Step 1: Set up the partial fraction decomposition. For distinct linear factors, the form is: (1)/((x-1)(x+2)) = (A)/(x-1) + (B)/(x+2) Step 2: Multiply both sides by the common denominator (x-1)(x+2). 1 = A(x+2) + B(x-1) Step 3: Solve for A and B. To find A, let x=1: 1 = A(1+2) + B(1-1) 1 = 3A A = (1)/(3) To find B, let x=-2: 1 = A(-2+2) + B(-2-1) 1 = -3B B = -(1)/(3) Step 4: Substitute the values of A and B back into the partial fraction form. (1)/((x-1)(x+2)) = (1)/(3)x-1 + (-1)/(3)x+2 = (1)/(3(x-1)) - (1)/(3(x+2)) The partial fraction decomposition is (1)/(3(x-1)) - (1)/(3(x+2)). Question 7: Split (3)/((x-2)^2(x+1)) into partial fractions. Step 1: Set up the partial fraction decomposition. For a repeated linear factor (x-2)^2 and a distinct linear factor (x+1), the form is: (3)/((x-2)^2(x+1)) = (A)/(x-2) + (B)/((x-2)^2) + (C)/(x+1) Step 2: Multiply both sides by the common denominator (x-2)^2(x+1). 3 = A(x-2)(x+1) + B(x+1) + C(x-2)^2 Step 3: Solve for A, B, and C. To find B, let x=2: 3 = A(2-2)(2+1) + B(2+1) + C(2-2)^2 3 = 0 + B(3) + 0 3 = 3B B = 1 To find C, let x=-1: 3 = A(-1-2)(-1+1) + B(-1+1) + C(-1-2)^2 3 = 0 + 0 + C(-3)^2 3 = 9C C = (3)/(9) = (1)/(3) To find A, substitute B=1, C=(1)/(3) and choose a convenient value for x, for example x=0: 3 = A(0-2)(0+1) +