Let's Simplify A Radical Quintic

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[@XJWill1]

Here is simpler solution. Using the notation in the video, the problem asks for a simplified x from x = a + b a = (sqrt(125)/2 + 11/2)^(1/5) b = (sqrt(125)/2 - 11/2)^(1/5) a*b = 1 so b = 1/a so: {1} x = a + 1/a {2} x^2 - 2 = a^2 + 1/a^2 x*(x^2 - 2) = (a + 1/a)*(a^2 + 1/a^2) x^3 - 2*x = a^3 + 1/a^3 + (a + 1/a) {3} x^3 - 3*x = a^3 + 1/a^3 , now multiply {2} * {3} {4} x^5 - 5*x^3 + 5*x = a^5 + 1/a^5 a^5 + 1/a^5 = a^5 + b^5 = sqrt(125) {4} x^5 - 5*x^3 + 5*x = sqrt(125) and square both sides {5} x^10 - 10*x^8 + 35*x^6 - 50*x^4 + 25*x^2 = 125 This is actually a quintic in x^2 , and the rational root theorem helps to find this factorization: (x^2 - 5)*(x^8 - 5*x^6 + 10*x^4 + 25) = 0 So x = sqrt(5) is a root, and that is the answer to the problem.

[@XJWill1]

By the way, it is worth noting that if the problem asked for the difference of the 5th roots instead of the sum, the math is slightly simpler and the answer is 1. If we define the sum and difference of the 5th roots as S and D, then S + D = 2*a S - D = 2/a (S + D)*(S - D) = 4 S^2 = 4 + D^2 S = sqrt(D^2 + 4) So the sum of the 5th roots is sqrt(1+4)

[@PeterMcDaid]

I looked at this and considered the expansion of (1+sqrt5)^5. Thinking of the rational component of the expansion gives 176. Dividing by 16 gives 11. So it becomes apparent that the first expression is the fifth root of {(1+sqrt5)/2}^5. The second expression is fifth root {(-1+sqrt5)^5}. Hence the sum =sqrt5. Perhaps intuitive but I was expecting a simple solution so looked for a simple approach.

[@shobithvasam3450]

Since ab=1, we can substitute b=1/a in (a^5)+(b^5)=5^(1/5) , and solve the quadratic for a^5 and find a

[@XJWill1]

Huh? That gets you nothing. We already know that a = (5*sqrt(5)/2 + 11/2)^(1/5)

[@shobithvasam3450]

Yeah oops forgot we were searching for a+b💀💀

[@carlosfabiano9171]

Using Excel calculation I got 2*(phi)-1 and this is sqrt(5) 😮

[@hkemal2743]

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