To simplify the notations, let a = (5-2*sqrt(13))^1/3, b = (5+2*sqrt(13))^1/3, then let x=a+b we get the cubic equation x^3+9x=10, obtaining x=1 as a solution. Do the same steps for y=a-b to get y^3-9y=-4*sqrt(13), with y=-sqrt(13) as a solution. Now we have a+b=1 and a-b=-sqrt(13) which are simply solved to get a = (1-sqrt(13))/2 = (5-2*sqrt(13))^1/3.
Although Wolfram Alpha agrees that (-5+2√13)^(1/3) = (-1+√13)/2, it does not agree that (5-2√13)^(1/3)= (1-√13)/2. It has to do with the definition of (cube) roots of negative numbers; the cube root of -1 is not commonly defined to be -1. And, just like √((-1)^2) is not equal to (-1), the same is true for ((-1)^3)^(1/3). The power rules which generally hold for integer powers also extend to arbitrary powers of positive real numbers, but they may fail for arbitrary powers of negative or complex numbers. Another example of what may go wrong is that √-4 * √-4 = (2i)*(2i)= -4, while √(-4)*(-4) =√16=4.
To simplify the notations, let a = (5-2*sqrt(13))^1/3, b = (5+2*sqrt(13))^1/3, then let x=a+b we get the cubic equation x^3+9x=10, obtaining x=1 as a solution. Do the same steps for y=a-b to get y^3-9y=-4*sqrt(13), with y=-sqrt(13) as a solution. Now we have a+b=1 and a-b=-sqrt(13) which are simply solved to get a = (1-sqrt(13))/2 = (5-2*sqrt(13))^1/3.
How do you get that cubic equation?
Thanks for sharing your insightful calculation! 💯💖🙏😎
@@KrytenKoro From x = a+b, we have x^3 = a^3+b^3 +3ab(a+b) = 5-2*sqrt(13) + 5+2*sqrt(13) + 3*{5^2-(2*sqrt(13)^2)}^1/3*x = 10-9x or x^3+9x=10.
Mui bueño diccion e didática.
😂❤
Maricá, Rio de janeiro
1-(13^(1/2)/2
8 , , Y- ..... 8
Although Wolfram Alpha agrees that
(-5+2√13)^(1/3) = (-1+√13)/2,
it does not agree that
(5-2√13)^(1/3)= (1-√13)/2.
It has to do with the definition of (cube) roots of negative numbers; the cube root of -1 is not commonly defined to be -1. And, just like √((-1)^2) is not equal to (-1), the same is true for ((-1)^3)^(1/3). The power rules which generally hold for integer powers also extend to arbitrary powers of positive real numbers, but they may fail for arbitrary powers of negative or complex numbers. Another example of what may go wrong is that
√-4 * √-4 = (2i)*(2i)= -4, while
√(-4)*(-4) =√16=4.
Thanks for your observation 💯💕✅🥰
it was necessary to explain what means + - square root, equall.
1.3-
у индусов странный английский
Which country are you from?
😅😂🤣Это, когда они живут в Кении! 😆😡😬