Simplify \(5\sqrt{18} - 3\sqrt{72} + 4\sqrt{50}\)

To simplify these expressions, we need to simplify the numbers under the radical sign first by factoring out their perfect square factors. For the first expression 5√18, we can factor out the perfect square factor of 9, which leaves us with 5√2√2√2 or 10√2. For the second expression, we can simplify √72 to √(36*2), and since 36 is a perfect square, we can factor it out, which leaves us with 3√2*6, or 3√2*2√3, which simplifies to 6√6. For the third expression, we can simplify √50 to √(25*2), and since 25 is a perfect square, we can factor it out, which leaves us with 2√2*5, or 2√2*√5, which simplifies to 2√10. Putting it all together, we have: 5√18 = 10√2 - 3√72 = 6√6 - 4√50 = 2√10 Now we can substitute these simplified expressions back into the original expressions: 10√2 - 6√6 + 2√10 To simplify this expression, we can group like terms. The coefficients of √6 are -6 and 0, since there are no other terms with √6. The coefficients of √10 are 2 and 0, since there are no other terms with √10. The coefficient of √2 is 10. So our simplified expression is: 10√2 - 6√6 + 2√10 = 10√2 - 6√6 + 2√10 = 10√2 - 6√(2*3) + 2√(2*5) = 10√2 - 6√2√3 + 2√2√5 = (10-6√3+2√5)√2 Therefore, the answer is 17√2.