To simplify radical expressions, we need to simplify the radicands, which are the numbers inside the square roots. We can simplify the radicands by finding the prime factors of each number and grouping them.

√18 = √(2 × 3 × 3) = 3√2
√20 = √(2 × 2 × 5) = 2√5
√8 = √(2 × 2 × 2) = 2√2
√45 = √(3 × 3 × 5) = 3√5


After simplifying the radicands, we can then combine like terms by adding or subtracting the coefficients (the numbers in front of the square roots) of the radicands that have the same radicand.

12√2 - 6√5 - 3(2√2) + √5(3√5)
= 12√2 - 6√5 - 6√2 + 3√25
= 6√2 - 3√5 + 15
= 6√2 - 3√5 + 15


The final answer can be simplified further by factoring out any common factors among the terms. In this case, there are no common factors among the terms, so the answer cannot be simplified further.

Therefore,

12√18 - 6√20 - 3√8 + √45
= 6√2 - 3√5 + 15

12√18-6√20-3√8-√45

=12×√9×√2 - 6×√4×√5 - 3×√4×√2 - √9×√5

=12×3×√2 - 6×2×√5 - 3×2×√2 - 3×√5

=36√2-12√5 - 6√2 - 3√5

=30√2 - 15√5(Ans).