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

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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

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3. Given vectors U = (2, 1) and V = (3, 2)