4:43 There must be a way of finding solutions to problems like this without so much work. ((√5−1)/2)^2 is ((5+1) -2(√5*1))/4 =(3 -√5)/2 in general ((a+b√5)/2)² = [(a²+5b²)/2 + ab√5]/2 note that if we start with a and b odd then aa and bb are odd thus the expression aa +5bb is even and so we will stay with integers so with that formula x²= ((a+b√5)/2)² = [(a²+5b²)/2 + ab√5]/2 we can set out a table power 1 a =-1 b =1 2 a
4:43 There must be a way of finding solutions to problems like this without so much work.
((√5−1)/2)^2 is ((5+1) -2(√5*1))/4 =(3 -√5)/2
in general
((a+b√5)/2)² =
[(a²+5b²)/2 + ab√5]/2
note that if we start with a and b odd then aa and bb are odd thus the expression aa +5bb is even and so we will stay with integers
so with that formula x²= ((a+b√5)/2)² = [(a²+5b²)/2 + ab√5]/2
we can set out a table
power 1 a =-1 b =1
2 a
Find the cube, then square the result 3 times. It's a lot easier.
@ 4:02 Once you rationalize the inside fraction... set a = (√5 - 1)/2
Then 2a + 1 = √5
4a² + 4a + 1 = 5
4a² + 4a - 4 = 0
a² + a - 1 = 0
So, a² = 1 - a .
[^^ This is a little "nicer" reduction than the x in your solution]
The rest:
a³ = a² * a = (1 - a)a = a - a² = a - (1-a) = 2a - 1
a⁶ = (a³)² = (2a - 1)² = 4a² - 4a + 1
= 4(1-a) - 4a + 1 = 5 - 8a
a^12 = (a⁶)² = (5 - 8a)² = 25 - 80a + 64a²
= 25 - 80a + 64(1-a)
= 89 - 144a
a^24 = (a^12)² = (89 - 144a)²
= 89² - 2(89)(144)a + 144²a²
= 7921 - 25632a + 20736(1-a)
= 28657 - 46368a
= 28657 - 46368 [(√5 - 1)/2 ]
...
= 51841 - 23184√5
Done!
a² = 1 - a is obvious, because the base is known as 1 / φ = φ - 1 (Golden Ratio). So we want φ^-24.
Yes ! Your method is easier to understand.Congratulation.
Hi
❤❤❤
Thanks 🚀🙏✅
Wow
1+1=?
set x=1
1+1=x+x=2x
so 1+1=2
This is not “simplify”, it makes it more complicate.
Are u sure without calculators???
En el 7'45
a.a.a = 3a
Y tú pusiste
a.a.a = 5a
a= raiz de 5
Peace be upon you ,it could be solved through logarithm , it would be easier !
Sehr umständlich. Aber geschaft . Gratuliere.
That's how it should be done ✅
calculate by reduce-algebra
(sqrt(24)/(sqrt(30)+sqrt(6)))^24;
1/(23184*sqrt(5)+51841)
Αbsolute stupid problem
0,46log24
The answer is zero. Try it
No it isn't. But you can round it to 0.00001 != 0.
wrong answer !
I've verified the answer. It's correct. I came to the same answer as in the video.
Не купуйтесь на велики цифри, відповідь при обчислюванні дорівнює 0,002
@AlreschaAndromeda The answer is also 1 / (103682 - 0.000009644875678449)! Picture this!
@AlreschaAndromeda: No, your number is merely an approximate.