Rationalize (√(3) - √(2)) / (√(5) - √(3)) and solve √(x² + 10x + 32) = 4

Let's solve these math problems step-by-step! 1. Rationalize (√(3) - √(2)) / (√(5) - √(3)) To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (√(5) - √(3)) is (√(5) + √(3)). Step 1: Multiply the numerator and denominator by the conjugate. (√(3) - √(2)) / (√(5) - √(3)) × (√(5) + √(3)) / (√(5) + √(3)) Step 2: Expand the numerator using the distributive property (FOIL). Numerator = (√(3) - √(2))(√(5) + √(3)) = √(3)·√(5) + √(3)·√(3) - √(2)·√(5) - √(2)·√(3) = √(15) + 3 - √(10) - √(6) Step 3: Expand the denominator using the difference of squares formula: (a - b)(a + b) = a² - b². Denominator = (√(5) - √(3))(√(5) + √(3)) = (√(5))² - (√(3))² = 5 - 3 = 2 Step 4: Combine the expanded numerator and denominator to get the rationalized expression. (√(15) + 3 - √(10) - √(6)) / 2 Answer: (√(15) + 3 - √(10) - √(6)) / 2 --- 2. Solve √(x² + 10x + 32) = 4 Step 1: Square both sides of the equation to eliminate the square root. (√(x² + 10x + 32))² = 4² x² + 10x + 32 = 16 Step 2: Move all terms to one side to form a standard quadratic equation (ax² + bx + c = 0). x² + 10x + 32 - 16 = 0 x² + 10x + 16 = 0 Step 3: Factor the quadratic equation. We need two numbers that multiply to 16 and add to 10. These numbers are 2 and 8. (x + 2)(x + 8) = 0 Step 4: Set each factor equal to zero and solve for x. x + 2 = 0 ⟹ x = -2 x + 8 = 0 ⟹ x = -8 Step 5: Check the solutions in the original equation. For x = -2: √( (-2)² + 10(-2) + 32 ) = √( 4 - 20 + 32 ) = √(16) = 4. (Correct) For x = -8: √( (-8)² + 10(-8) + 32 ) = √( 64 - 80 + 32 ) = √(16) = 4. (Correct) Answer: x = -2 or x = -8 --- 3. Given vectors U = (2, 1) and V = (3, 2) c) If xU + yV = (1, 0), find x and y. Step 1: Substitute the vector components into the given equation. x(2, 1) + y(3, 2) = (1, 0) Step 2: Perform scalar multiplication for each vector. (2x, x) + (3y, 2y) = (1, 0) Step 3: Add the corresponding components of the vectors. (2x + 3y, x + 2y) = (1, 0) Step 4: Equate the corresponding components to form a system of linear equations. 1) 2x + 3y = 1 2) x + 2y = 0 Step 5: Solve the system of equations. From equation (2), we can express x in terms of y: x = -2y Step 6: Substitute x = -2y into equation (1). 2(-2y) + 3y = 1 -4y + 3y = 1 -y = 1 y = -1 Step 7: Substitute the value of y back into x = -2y to find x. x = -2(-1) x = 2 Answer: x = 2 and y = -1 d) Find the modulus of U. The modulus (or magnitude) of a vector