Surds (addition, subtraction, multiplying and dividing)

Multiply the coefficients (4 * 2 = 8) and the radicands (sqrt(15) * sqrt(18) = sqrt(270)). Simplify sqrt(270) = sqrt(9 * 30) = 3 * sqrt(30). Thus, 8 * 3 * sqrt(30) = 24 * sqrt(30).

Divide the coefficients (-8/2 = -4) and the radicals (sqrt(2)/sqrt(26) = sqrt(2/26) = sqrt(1/13)). The result is -4 * sqrt(1/13).

Distribute the sqrt(3) to both terms inside the parentheses: sqrt(3) * sqrt(2) = sqrt(6) and sqrt(3) * sqrt(5) = sqrt(15). The expression becomes sqrt(6) - sqrt(15).

To express a coefficient as part of a square root, square the coefficient and multiply it by the radicand: 11 * sqrt(3) = sqrt(11^2 * 3) = sqrt(121 * 3) = sqrt(363).

Simplify the radicals first: sqrt(50) = sqrt(25 * 2) = 5 * sqrt(2) and sqrt(8) = sqrt(4 * 2) = 2 * sqrt(2). Then, 5 * sqrt(2) - 2 * sqrt(2) = 3 * sqrt(2).

Find a common denominator for the fractions 6 and 8, which is 24. The expression becomes (-20 * sqrt(10) / 24) + (9 * sqrt(10) / 24) = -11 * sqrt(10) / 24.

Simplify sqrt(147) = sqrt(49 * 3) = 7 * sqrt(3). The expression becomes (3 * 7 * sqrt(3)) / 14 = 21 * sqrt(3) / 14. Dividing both by 7 gives (3 * sqrt(3)) / 2.

Group the like terms: (2 * sqrt(3) + 2 * sqrt(3)) + (2 * sqrt(7) - 2 * sqrt(7)) = 4 * sqrt(3) + 0 = 4 * sqrt(3).

sqrt(45) = sqrt(9 * 5) = sqrt(9) * sqrt(5) = 3 * sqrt(5).