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x³ = (x⁶ + 1)/702 → you can see that: x³ > 0 → x > 0a = x² + (1/x⁵) → we've just seen that: x > 0 → a > 0x⁵ = (x¹º + 1)/aa = (x¹º + 1)/x⁵a = (x¹º/x⁵) + (1/x⁵)a = x⁵ + (1/x⁵) ← memorize this resultRestartx³ = (x⁶ + 1)/702x⁶ + 1 = 702x³x⁶ - 702x³ + 1 = 0Δ = (- 702)² - (4 * 1) = 492800 = 80² * 77x³ = (702 ± 80√77)/2x³ = 351 ± 40√77First case: x³ = 351 + 40√77 → suppose that:y³ = 351 - 40√77-----------------------------------------------------subtractionx³ + y³ = 702x³y³ = (351 + 40√77).(351 - 40√77)x³y³ = 351² - (40² * 77)x³y³ = 1xy = 1Identity:x³ + y³ = (x + y).(x² - xy + y²) → recall: x³ + y³ = 702(x + y).(x² - xy + y²) = 702(x + y).[x² + y² - xy] = 702(x + y).[(x + y)² - 2xy - xy] = 702(x + y).[(x + y)² - 3xy] = 702 → recall: xy = 1(x + y).[(x + y)² - 3] = 702 → let: m = x + ym.(m² - 3) = 702m³ - 3m - 702 = 0 → 702 = 729 - 27 = 9³ - (3 * 9) ← you can see that m = 3 is an obvious solutionm = 9 → recall: m = x + yx + y = 9 → recall: xy = 1 → y = 1/xx + (1/x) = 9(x² + 1)/x = 9x² + 1 = 9xx² - 9x + 1 = 0Δ = (- 9)² - (4 * 1) = 81 - 4 = 77x = (9 ± √77)/2First case: x = (9 + √77)/2x² = [(9 + √77)/2]²x² = (9 + √77)²/2²x² = (81 + 18√77 + 77)/2²x² = (158 + 18√77)/2²x² = (79 + 9√77)/2x⁴ = [(79 + 9√77)/2]²x⁴ = (79 + 9√77)²/2²x⁴ = (6241 + 1422√77 + 6237)/2²x⁴ = (12478 + 1422√77)/2²x⁴ = (6239 + 711√77)/2x⁵ = x⁴.xx⁵ = [(6239 + 711√77)/2].(9 + √77)/2x⁵ = (6239 + 711√77).(9 + √77)/41/x⁵ = 4/[(6239 + 711√77).(9 + √77)]1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(6239 + 711√77).(6239 - 711√77).(9 + √77).(9 - √77)]1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(6239² - {711² * 77}).(81 - 77)]1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(4).(4)]1/x⁵ = (6239 - 711√77).(9 - √77)/41/x⁵ = (56151 - 6239√77 - 6399√77 + 54747)/41/x⁵ = (110898 - 12638√77)/41/x⁵ = (55449 - 6319√77)/2Recall the memorized resulta = x⁵ + (1/x⁵)a = [(6239 + 711√77).(9 + √77)/4] + [(55449 - 6319√77)/2]a = [(6239 + 711√77).(9 + √77) + 2.(55449 - 6319√77)]/4a = [56151 + 6239√77 + 6399√77 + 54747 + 110898 - 12638√77]/4a = (56151 + 54747 + 110898)/4a = 221796/4a = 55449 ← you can see that does not depend on √77Second case: x = (9 - √77)/2 → a = 55449
55449
x³ = (x⁶ + 1)/702 → you can see that: x³ > 0 → x > 0
a = x² + (1/x⁵) → we've just seen that: x > 0 → a > 0
x⁵ = (x¹º + 1)/a
a = (x¹º + 1)/x⁵
a = (x¹º/x⁵) + (1/x⁵)
a = x⁵ + (1/x⁵) ← memorize this result
Restart
x³ = (x⁶ + 1)/702
x⁶ + 1 = 702x³
x⁶ - 702x³ + 1 = 0
Δ = (- 702)² - (4 * 1) = 492800 = 80² * 77
x³ = (702 ± 80√77)/2
x³ = 351 ± 40√77
First case:
x³ = 351 + 40√77 → suppose that:
y³ = 351 - 40√77
-----------------------------------------------------subtraction
x³ + y³ = 702
x³y³ = (351 + 40√77).(351 - 40√77)
x³y³ = 351² - (40² * 77)
x³y³ = 1
xy = 1
Identity:
x³ + y³ = (x + y).(x² - xy + y²) → recall: x³ + y³ = 702
(x + y).(x² - xy + y²) = 702
(x + y).[x² + y² - xy] = 702
(x + y).[(x + y)² - 2xy - xy] = 702
(x + y).[(x + y)² - 3xy] = 702 → recall: xy = 1
(x + y).[(x + y)² - 3] = 702 → let: m = x + y
m.(m² - 3) = 702
m³ - 3m - 702 = 0 → 702 = 729 - 27 = 9³ - (3 * 9) ← you can see that m = 3 is an obvious solution
m = 9 → recall: m = x + y
x + y = 9 → recall: xy = 1 → y = 1/x
x + (1/x) = 9
(x² + 1)/x = 9
x² + 1 = 9x
x² - 9x + 1 = 0
Δ = (- 9)² - (4 * 1) = 81 - 4 = 77
x = (9 ± √77)/2
First case: x = (9 + √77)/2
x² = [(9 + √77)/2]²
x² = (9 + √77)²/2²
x² = (81 + 18√77 + 77)/2²
x² = (158 + 18√77)/2²
x² = (79 + 9√77)/2
x⁴ = [(79 + 9√77)/2]²
x⁴ = (79 + 9√77)²/2²
x⁴ = (6241 + 1422√77 + 6237)/2²
x⁴ = (12478 + 1422√77)/2²
x⁴ = (6239 + 711√77)/2
x⁵ = x⁴.x
x⁵ = [(6239 + 711√77)/2].(9 + √77)/2
x⁵ = (6239 + 711√77).(9 + √77)/4
1/x⁵ = 4/[(6239 + 711√77).(9 + √77)]
1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(6239 + 711√77).(6239 - 711√77).(9 + √77).(9 - √77)]
1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(6239² - {711² * 77}).(81 - 77)]
1/x⁵ = 4.(6239 - 711√77).(9 - √77) / [(4).(4)]
1/x⁵ = (6239 - 711√77).(9 - √77)/4
1/x⁵ = (56151 - 6239√77 - 6399√77 + 54747)/4
1/x⁵ = (110898 - 12638√77)/4
1/x⁵ = (55449 - 6319√77)/2
Recall the memorized result
a = x⁵ + (1/x⁵)
a = [(6239 + 711√77).(9 + √77)/4] + [(55449 - 6319√77)/2]
a = [(6239 + 711√77).(9 + √77) + 2.(55449 - 6319√77)]/4
a = [56151 + 6239√77 + 6399√77 + 54747 + 110898 - 12638√77]/4
a = (56151 + 54747 + 110898)/4
a = 221796/4
a = 55449 ← you can see that does not depend on √77
Second case: x = (9 - √77)/2 → a = 55449
55449