Nice Math Olympiad Simplification | Denesting 101 | Algebraic Approach

[@rainerzufall42]

I've tried your strange method (sorting roots and integers) on the original equation. Strangely it worked out! qrt(56 - 24 sqrt(5)) = sqrt(a) - sqrt(b) a^2 - 4 sqrt(a) a sqrt(b) + 6 ab - 4 sqrt(a) b sqrt(b) + b^2 = 56 - 24 sqrt(5) Sorting for integers: a^2 + 6 ab + b^2 = 56 Sorting for roots: -4(a+b) sqrt(a) sqrt(b) = -24 sqrt(5) => (a+b)^2 * ab = 6^2 * 5 = 180 => (a^2 + 2 ab + b^2) ab = 180 Use the integer formula from above for 56 - 4 ab: (56 - 4 ab) ab = 180 Substitute x = ab: 4 x^2 - 56 x + 180 = 0 => x^2 - 14 x + 45 = 0 => x = 7 +/- sqrt(49 - 45) = { 5, 9 } = ab

[@rainerzufall42]

CASE ab = 5: (a+b)^2 = 180 / ab = 180/5 = 36 => a + b = +/- 6 (a, b) = (5, 1) or (a, b) = (-5, -1) < (0, 0), rejected CASE ab = 9: (a+b)^2 = 180 / ab = 180/9 = 20 => a + b = +/- 2 sqrt(5) (a+9/a) = +/- sqrt(20) a^2 -/+ sqrt(20) a + 9 = 0 delta = 20 - 4 * 9 = -16 < 0, no real roots Thus a = 5, b = 1: qrt(56 - 24 sqrt(5)) = sqrt(5) - sqrt(1) = sqrt(5) - 1

[@rainerzufall42]

CASE ab = 9: (a+b)^2 = 180 / ab = 180/9 = 20 => a + b = +/- 2 sqrt(5) (a+9/a) = +/- sqrt(20) a^2 -/+ sqrt(20) a + 9 = 0 delta = 20 - 4 * 9 = -16 < 0, no real roots

[@rainerzufall42]

Thus a = 5, b = 1: qrt(56 - 24 sqrt(5)) = sqrt(5) - sqrt(1) = sqrt(5) - 1

[@NadiehFan]

Much too longwinded. For denesting nested _square_ roots √(a ± √b) with a, b > 0, a² − b > 0 we can use the identity which says that if c = √(a² − b) then √(a ± √b) = √((a + c)/2) ± √((a − c)/2) First we denest √(56 − 24√5) where we can take out a factor 4 = 2² and write this as 2√(14 − 6√5) = 2√(14 − √180) to ease calculations. Here a = 14, b = 180 so c = √(196 − 180) = √16 = 4 and we have √(56 − 24√5) = 2√(14 − √180) = 2(√9 − √5) = 2(3 − √5) = 6 − 2√5 Next we denest √(6 − 2√5) = √(6 − √20) where a = 6, b = 20 so c = √(36 − 20) = √16 = 4 and we have √(6 − 2√5) = √5 − √1 = √5 − 1 So, we have ⁴√(56 − 24√5) = √(√(56 − 24√5)) = √(6 − 2√5) = √5 − 1

[@superacademy247]

Thanks 👍💯 for your input. Well explained.

[@vladimirberman9953]

To start, it's important to clearly define what you mean by "simplify an expression." For example, if by simplifying you mean expressing the result as 𝐴 +𝐵*sqrt(5) , where A and 𝐵B are integers, then consider raising the expression to the fourth power. This leads to a set of two nonlinear equations (for A and B), if you're looking for integer solutions. If, however, you are considering solutions within the realm of real numbers, then there are infinitely many solutions.

[@rainerzufall42]

CASE ab = 5: (a+b)^2 = 180 / ab = 180/5 = 36 => a + b = +/- 6 (a, b) = (5, 1) or (a, b) = (-5, -1) < (0, 0), rejected

[@vladimirberman9953]

A=-1 and B=1

[@rainerzufall42]

CASE ab = 5: (a+b)^2 = 180 / ab = 180/5 = 36 => a + b = +/- 6 (a, b) = (5, 1) or (a, b) = (-5, -1) < (0, 0), rejected