Multiply top and bottom by sqrt(10). This gives (5 sqrt(10)) / 10. Simplify the fraction by dividing numerator and denominator by 5, resulting in sqrt(10) / 2.

Multiply by the conjugate (sqrt(3) + 1). The denominator evaluates to (sqrt(3))^2 - 1^2 = 3 - 1 = 2. The expression becomes 2(sqrt(3) + 1) / 2. Canceling the 2s leaves sqrt(3) + 1.

Multiply by the conjugate, (sqrt(5) + sqrt(2)). The new denominator is (sqrt(5))^2 - (sqrt(2))^2 = 5 - 2 = 3. The numerator is 1 * (sqrt(5) + sqrt(2)). The result is (sqrt(5) + sqrt(2)) / 3.

Multiply the numerator and denominator by the conjugate of the denominator, which is (2 - sqrt(3)). The denominator becomes (2^2 - (sqrt(3))^2) = 4 - 3 = 1. The result is just the numerator: 2 - sqrt(3).

Multiply numerator and denominator by sqrt(5). This results in (10 sqrt(5)) / 5. Divide 10 by 5 to get the final simplified form: 2 sqrt(5).

Multiply the top and bottom by sqrt(3). The expression becomes (3 * sqrt(3)) / (sqrt(3) * sqrt(3)) = (3 sqrt(3)) / 3. The 3s cancel out, leaving just sqrt(3).

To remove the surd from the denominator, multiply the numerator and the denominator by sqrt(2). This gives (1 * sqrt(2)) / (sqrt(2) * sqrt(2)) = sqrt(2) / 2.

Combine under a single root: sqrt(18 / 2). This equals sqrt(9). The square root of 9 evaluates to 3.

Square both parts of the term separately. The square of 2 is 4, and the square of sqrt(5) is 5. Multiply the results: 4 * 5 = 20.

Divide the outside coefficients: 10 / 2 = 5. Divide the terms inside the roots: sqrt(10) / sqrt(2) = sqrt(5). Combining them yields 5 sqrt(5).