For x^n you just need to look at the nth term of two different Fibonacci series with alternating plus/minus signs, divided by 2. -1,3,-4,7,-11,18,-29,,, and 1,-1,2,-3,5,-8.13,,,, ( -29 + 13*sqrt(5) ) / 2
The given expression for x simplifies to x = [√3 -1]/2 + [√5 - √3 ]/2 = [√5 -1]/2. This is a solution of x^2+x-1=0. So, x^2=-x+1. This can be iterated to yield x^7 = 13x-8 = [13√5 -29]/2.
For x^n you just need to look at the nth term of two different Fibonacci series with alternating plus/minus signs, divided by 2. -1,3,-4,7,-11,18,-29,,, and 1,-1,2,-3,5,-8.13,,,, ( -29 + 13*sqrt(5) ) / 2
It was a wonderful introduction....thanks Sir 🙏 for sharing
The given expression for x simplifies to x = [√3 -1]/2 + [√5 - √3 ]/2 = [√5 -1]/2. This is a solution of x^2+x-1=0. So, x^2=-x+1. This can be iterated to yield x^7 = 13x-8 = [13√5 -29]/2.
x simplifies to 1/2(sqrt5-1)
then x^7=1/2{13sqrt5-29}
x^7=(13√5-29) /2
8√5-22
A Nice Radical Math Problem:
x = (√6 + 1)/(√6 + √3 + 3√2 + 1) + (√5 + 1)/(√15 + 5 + √5 + √3), x⁷ = ?
√6 + 1 + 3√2 + √3 = (√6 + 1) + (√3)(√6 + 1) = (√6 + 1)(√3 + 1)
5 + √15 + √5 + √3 = √5(√5 + √3) + (√5 + √3) = (√5 + 1)(√5 + √3)
x = 1/(√3 + 1) + 1/(√5 + √3) = (√3 - 1)/(3 - 1) + (√5 - √3)/(5 - 3)
= (√3 - 1)/2 + (√5 - √3)/2 = (√5 - 1)/2
x² = [(√5 - 1)/2]² = (5 + 1 - 2√5)/4 = (3 - √5)/2 = 1 - (√5 - 1)/2 = 1 - x
x³ = x(x²) = x(1 - x) = x - x² = x - (1 - x) = 2x - 1
x⁴ = (1 - x)² = 1 - 2x + x² = 1 - 2x + (1 - x) = 2 - 3x
x⁷ = (x³)(x⁴) = (2x - 1)(2 - 3x) = 7x - 2 - 6x² = 7x - 2 - 6(1 - x) = 13x - 8
= 13[(√5 - 1)/2] - 8 = (13√5 - 13 - 16)/2 = (13√5 - 29)/2
(-29+13×ριζα5)/2
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{6x+6x ➖ }{1x+1x ➖ }/{6x+6x ➖}{3x+3x ➖3x+3x ➖ {2x+2x ➖1x+1x ➖}= {12x^2+2x^2}/ {6x^2+6x^2}{4x^2+2x^2} =14x^2/{12x^4+6x^4}= 14x^2/18x^8=1.4x^3 1^1.2^2x^3^1 1^2x^3^1 2x^3 (x ➖ 3x+2).{5x+5x ➖ 1x+1x ➖}/{15x+15x ➖ 5x+5x ➖}{5x+5x ➖3x+3x ➖}= 12x^4/{40x^4+16x^4} = 12x^4/56x^8= 4.8x^8 2^2.2^3x^2^3,1^1.2^1x^1^1 2x^1 (x ➖ 2x+1) .
√6 + √3 + 3√2 + 1
= √6(√3 + 1) + (√3 + 1)
= (√6 + 1)(√3 + 1)
(√6 + 1)/(√6 + √3 + 3√2 + 1) = 1/(√3 + 1)
√15 + 5 + √5 + √3
= √5(√3 + √5) + (√3 + √5)
= (√5 + 1)(√5 + √3)
(√5 + 1)/(√15 + 5 + √5 + √3) = 1/(√5 + √3)
x = 1/(√3 + 1) + 1/(√5 + √3)
x = (√3 - 1)/2 + (√5 - √3)/2
x = (√5 - 1)/2
x² = (3 - √5)/2
x⁴ = (7 - 3√5)/2
x⁶ = (18 - 8√5)/2
x⁷ = (18 - 8√5)(√5 - 1)/4
*x⁷ = (13√5 - 29)/2*
x= (13√5-29)/2