Master Radical Math with This ONE Simple Technique!

[@Quest3669]

X^2= (3-√5)/2 & x= (√5-1)/2 hence x^17=(1597√5-3571)/2

[@michaeledwards2251]

My approach would be 17 = 10 + 5 + 2.
x^10 = (2 ^ 5) / ( 3 + root(5) ) ^5
x^2 = 2 / ( 3 + root(5) )
x^5 =(2 ^ 2.5) / ( 3 + root(5) ) ^ 2.5
x^17 =(2 ^ 8.5) / ( 3 + root(5) )^ 8.5 = 256 root(2) / (( 3 + root(5) ) ^ 8.5)
Given 2 and 5 are irrational roots, I would use a calculator, to answer to a specified number of digits.

[@RashmiRay-c1y]

x simplifies to x=1/2(√5-1). Thus, x^17 = 1/2(1597√5 -3571).

[@gregevgeni1864]

(-3571+1597√5)/2

[@RashmiRay-c1y]

@@gregevgeni1864 Thanks! I mistyped, forgetting to hit the "shift" key.

[@gregevgeni1864]

(3+√5)(3-√5) = 3² -√5²= 9-5=4 (#)
Then
x¹⁰ = [2/(3+√5)]⁵ (1) => x² = 2/(3+√5)= (3-√5)/2 = [(-1+√5)/2]² =>
x = (-1+√5)/2 (2).
x⁶ =(x²)³ = [(3-√5)/2]³ (3)
Then, due to (1), (2), (3) =>
x¹⁷ = x¹⁰ • x⁶ • x = [2/(3+√5)]⁵•[(3-√5)/2]³ •(-1+√5)/2 = 32/(3+√5)⁵ • (3-√5)³/8 •(-1+√5)/2 = [4(3-√5)³/(3+√5)⁵]•(-1+√5)/2 =
= {[(3-√5)(3+√5)(3-√5)³]/(3+√5)⁵}• (-1+√5)/2 = [(3-√5)/(3+√5)]⁴ •(-1+√5)/2
= [(3-√5)²/(3+√5)((3-√5)]⁴•(-1+√5)/2
= [(3-√5)²/4]⁴ •(-1+√5)/2 =
=[(7-3√5)/2]⁴ •(-1+√5)/2
=[(17656-7896√5)/16]•(-1+√5)/2
= (4414-1974√5)/4 •(-1+√5)/2
= .. = (-3571+1597√5)/2

[@ManojkantSamal]

^=read as to the power
*=read as square root
As per question
X^10=32/(3+*5)^5=2^5/(3+*5)^5
={2/(3+*5)}^5
(X^2)^5={2/(3+*5)}^5
X^2=2/(3+*5)={2(3-*5)/(3+*5)(3-*5)}
X^2={2(3-*5)/(9-5)}
={2(3-*5)/4}=(3-*5)/2.......EQN1
Now explain
(3-*5)/2=2(3-*5)/4
=(6-2.*5)/4
={(*5)^2+1^2-(2×1×*5)}/2^2
={(*5-1)/2}^2
So,
X^2 ={(*5-1)/2}^2
X=(*5-1)/2.........eqn2
Take the square of eqn1
(X^2)^2 ={(3-*5)/2}^2
X^4={9+5-(2×3×*5)}/4
=(14-6.*5)/4=2(7-3.*5)/4
=(7 - 3.*5)/2........eqn3
Take the fourth of eqn3
(X^4)^4=(7-3.*5)^4/2^4
X^16=(7-3.*5)^4/16
Let's explain N
N=(7-3.*5)^4
Let
Q=7, R=(-3.*5)
Q^4=7^4=2401
R^4=(-3.*5)^4=2025
Q^2=7^2=49
R^2=(-3.*5)^2=45
Q×R=7×(-3.*5)=(-21.*5)
6Q^2R^2=6×49×45=13230
According to the formula
(Q+R)^4=Q^4+R^4+6Q^2R^2+4QR(Q^2+R^2)
SO,
N=2401+2025 +13230+{(4×(-21.*5)}{49+45}
N=4426+13230+{-84.*5(94)}
N=17656-7896.*5
N/D=(17656-7896.*5)/16
=8(2207-987.*5)/16
=(2207-987.*5)/2.........EQN4
Eqn2 ×eqn4
(X^16)×X={(2207-987.*5)/2}×{(*5-1)/2}
N=(*5-1)(2207-987.*5)
=2207.*5-4935-2207+987.*5
=3194.*5-7142
=2(1597.*5-3571)
D=2×2=4
N/D=2(1597.*5 - 3571)/4
=(1597.*5 - 3571)/2
Hence
X^17=(1597.*5 - 3571)/2

[@에스피-z2g]

x^2=2/(3+rt5)
x=(rt5-1)/2
2x+1=rt5
x^2+x-1=0
x^2=1-x
x^4=(1-x)^2=2-3x
x^8=(2-3x)^2=
4-12x+9x^2=
13-21x
x^16=(13-21x)^2=
610-987x
x^17=610x-987x^2=
1597x-987=
(1597rt5-3571)/2

[@key_board_x]

Other way….
x¹⁰ = 32/(3 + √5)⁵
x¹⁰ = [2/(3 + √5)]⁵
(x²)⁵ = [2/(3 + √5)]⁵
x² = 2/(3 + √5)
x² = 2.(3 - √5)/[(3 + √5).(3 - √5)]
x² = 2.(3 - √5)/[9 - 5]
x² = (6 - 2√5)/4
x² = [1 - 2√5 + 5]/4
x² = [(1)² - 2.(1 * √5) + (√5)²]/2²
x² = (1 - √5)²/2²
x = ± (1 - √5)/2 → you know that: (1 - √5) < 0 → but recall the condition: x > 0
x = - (1 - √5)/2
x = (√5 - 1)/2
x³ = x² * x → recall: x² = (6 - 2√5)/4
x³ = [(6 - 2√5)/4] * x → recall: x = (√5 - 1)/2
x³ = [(6 - 2√5)/4].(√5 - 1)/2
x³ = [(3 - √5)/2].(√5 - 1)/2
x³ = (3 - √5).(√5 - 1)/4
x³ = (3√5 - 3 - 5 + √5)/4
x³ = (4√5 - 8)/4
x³ = √5 - 2
(3 + √5)² = 9 + 6√5 + 5
(3 + √5)² = 14 + 6√5
(3 + √5)² = 2.(7 + 3√5)
(3 + √5)⁴ = [(3 + √5)²]²
(3 + √5)⁴ = [2.(7 + 3√5)]²
(3 + √5)⁴ = 4.(7 + 3√5)²
(3 + √5)⁴ = 4.(49 + 42√5 + 45)
(3 + √5)⁴ = 4.(94 + 42√5)
(3 + √5)⁴ = 8.(47 + 21√5)
(3 + √5)⁵ = (3 + √5)⁴.(3 + √5)
(3 + √5)⁵ = 8.(47 + 21√5).(3 + √5)
(3 + √5)⁵ = 8.(141 + 47√5 + 63√5 + 105)
(3 + √5)⁵ = 8.(246 + 110√5)
(3 + √5)⁵ = 16.(123 + 55√5)
x¹⁰ = 32/(3 + √5)⁵ → recall the previous result
x¹⁰ = 32/[16.(123 + 55√5)]
x¹⁰ = 2/(123 + 55√5)
x¹⁰ = 2.(123 - 55√5)/[(123 + 55√5).(123 - 55√5)]
x¹⁰ = 2.(123 - 55√5)/[123² - (55√5)²]
x¹⁰ = 2.(123 - 55√5)/[15129 - 15125]
x¹⁰ = 2.(123 - 55√5)/[4]
x¹⁰ = (123 - 55√5)/2
x²⁰ = (x¹⁰)²
x²⁰ = [(123 - 55√5)/2]²
x²⁰ = [123 - 55√5]²/4
x²⁰ = [123² - 2.(123 * 55√5) + (55√5)²]/4
x²⁰ = [15129 - 13530√5 + 15125]/4
x²⁰ = [30254 - 13530√5]/4
x²⁰ = (15127 - 6765√5)/2
x¹⁷ = x²⁰/x³
x¹⁷ = [(15127 - 6765√5)/2] / x³ → recall: x³ = √5 - 2
x¹⁷ = [(15127 - 6765√5)/2] / (√5 - 2)
x¹⁷ = [(15127 - 6765√5).(√5 + 2)/2] / [(√5 - 2).(√5 + 2)]
x¹⁷ = [(15127 - 6765√5).(√5 + 2)/2] / [5 - 4]
x¹⁷ = (15127 - 6765√5).(√5 + 2)/2
x¹⁷ = (15127√5 + 30254 - 33825 - 13530√5)/2
x¹⁷ = (1597√5 - 3571)/2

[@潘博宇-k4l]

E=X^17=[3815(5)^(1/2)-7097]/4