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Compute $$\sqrt[3]{ \frac{80}{27} +\sqrt[3]{ \frac{80}{27}+\sqrt[3]{ \frac{80}{27} +... }}}$$

Jul 5, 2020

#1
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c=0;a=listfor(n, 1, 15, (listforeach(b,(80/27), c=3#(b + c))

OUPUT = (1.436289793, 1.638549773, 1.663285786, 1.666260862, 1.666617969, 1.666660823, 1.666665965, 1.666666583, 1.666666657, 1.666666665, 1.666666667, 1.666666667).

Note: As you can see, it converges very rapidly to = 1 2/3

Jul 5, 2020
#2
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You can solve it algebraically this way:

x = (80/27)^1/3 + x^1/3

Raise both sides to the 3rd power

x^3 =(80/27)  + x

x^3 - x - 80/27 = 0

Solve for x:
x^3 - x - 80/27 = 0

Factor the left hand side.
The left hand side factors into a product with three terms:
1/27 (3 x - 5) (9 x^2 + 15 x + 16) = 0

Multiply both sides by a constant to simplify the equation.
Multiply both sides by 27:
(3 x - 5) (9 x^2 + 15 x + 16) = 0

Find the roots of each term in the product separately.
Split into two equations:
3 x - 5 = 0 or 9 x^2 + 15 x + 16 = 0

Look at the first equation: Isolate terms with x to the left hand side.
3 x = 5 or 9 x^2 + 15 x + 16 = 0

Solve for x.
Divide both sides by 3:
x = 5/3 or 9 x^2 + 15 x + 16 = 0

Look at the second equation: Write the quadratic equation in standard form.
Divide both sides by 9:
x = 5/3 or x^2 + (5 x)/3 + 16/9 = 0

Solve the quadratic equation by completing the square.
Subtract 16/9 from both sides:
x = 5/3 or x^2 + (5 x)/3 = -16/9

Take one half of the coefficient of x and square it, then add it to both sides.
x = 5/3 or x^2 + (5 x)/3 + 25/36 = -13/12

Factor the left hand side.
Write the left hand side as a square:
x = 5/3 or (x + 5/6)^2 = -13/12

Eliminate the exponent on the left hand side.
Take the square root of both sides:
x = 5/3 or x + 5/6 = 1/2 i sqrt(13/3) or x + 5/6 = -1/2 i sqrt(13/3)

Look at the second equation: Solve for x.
Subtract 5/6 from both sides:
x = 5/3 or x = 1/2 i sqrt(13/3) - 5/6 or x + 5/6 = -1/2 i sqrt(13/3)

Look at the third equation: Solve for x.
Subtract 5/6 from both sides:
x = 5/3                                                                                                                                                                                 [or x = 1/2 i sqrt(13/3) - 5/6 or x = -1/2 i sqrt(13/3) - 5/6] - Discard these solutions.

Jul 5, 2020
edited by Guest  Jul 5, 2020