Manipulation of surds

Expanding (1 - sqrt(2))^2 gives (1 - sqrt(2))(1 - sqrt(2)) = 1 - sqrt(2) - sqrt(2) + (sqrt(2))^2 = 1 - 2sqrt(2) + 2 = 3 - 2sqrt(2).

Expanding (2 + sqrt(5))(3 - sqrt(5)) using FOIL: 23 - 2sqrt(5) + 3*sqrt(5) - (sqrt(5))^2 = 6 + sqrt(5) - 5 = 1 + sqrt(5).

Expand the expression using the FOIL method: (2 * 1) + (2 * sqrt(3)) + (sqrt(3) * 1) + (sqrt(3) * sqrt(3)). This simplifies to 2 + 2sqrt(3) + sqrt(3) + 3, which combines to 5 + 3sqrt(3).

Simplify each radical: sqrt(20) = 2sqrt(5), sqrt(180) = 6sqrt(5), and sqrt(125) = 5sqrt(5). Combining these gives 2sqrt(5) + 6sqrt(5) - 5sqrt(5) = 3*sqrt(5).

Simplify the radicals: sqrt(48) = sqrt(16 * 3) = 4sqrt(3) and sqrt(12) = sqrt(4 * 3) = 2sqrt(3). Subtracting them gives 4sqrt(3) - 2sqrt(3) = 2*sqrt(3).

Simplify the radicals: sqrt(18) = 3sqrt(2) and sqrt(8) = 2sqrt(2). Adding them gives 3sqrt(2) + 2sqrt(2) = 5*sqrt(2).

The square root of a difference is not equal to the difference of the square roots. For example, if a=9 and b=4, sqrt(9-4) = sqrt(5), while sqrt(9)-sqrt(4) = 3-2 = 1. Since sqrt(5) is not equal to 1, the statement is false.

The square root of a sum is not equal to the sum of the square roots. For example, if a=9 and b=16, sqrt(9+16) = sqrt(25) = 5, while sqrt(9)+sqrt(16) = 3+4 = 7. Since 5 is not equal to 7, the statement is false.

When squaring a fraction, square both the numerator and the denominator. (sqrt(3))^2 = 3 and 2^2 = 4, resulting in 3/4.

The expression is (asqrt(b) * csqrt(b)) / (asqrt(b)). You can cancel the common factor (asqrt(b)) from the numerator and denominator, which leaves c*sqrt(b).

Multiply the coefficients (a * c) and the radicals (sqrt(b) * sqrt(b)). Since sqrt(b) * sqrt(b) = b, the expression becomes a * c * b, or abc.

The expression is (4sqrt(2) * sqrt(3)) / (2sqrt(2)). Cancel the common factor sqrt(2) to get (4 * sqrt(3)) / 2, which simplifies to 2*sqrt(3).

Divide the coefficients (12 / 6 = 2) and the radicals (sqrt(56) / sqrt(8) = sqrt(56/8) = sqrt(7)). The result is 2*sqrt(7).

Divide the coefficients (3 / 1 = 3) and the radicals (sqrt(15) / sqrt(3) = sqrt(15/3) = sqrt(5)). The result is 3*sqrt(5).

Multiply the coefficients (2 * 3 = 6) and the radicals (sqrt(48) * sqrt(8) = sqrt(384)). Simplify sqrt(384) = sqrt(64 * 6) = 8sqrt(6). Then 6 * 8sqrt(6) = 48*sqrt(6).

Multiply the coefficients (2 * 3 = 6) and the radicals (sqrt(18) * sqrt(2) = sqrt(36)). Since sqrt(36) = 6, the expression becomes 6 * 6 = 36.

The expression is (sqrt(2) * sqrt(3)) / sqrt(2). The sqrt(2) terms cancel out, leaving sqrt(3).

sqrt(12) / sqrt(3) = sqrt(12 / 3) = sqrt(4).

The product of two square roots is the square root of the product: sqrt(a) * sqrt(b) = sqrt(a * b). Thus, sqrt(2) * sqrt(3) = sqrt(6).