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# Find all solutions so that ​ is true

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Find all solutions so that  $$\sqrt[3]{x + 57} - \sqrt[3]{x - 57} = \sqrt[3]{6}$$ is true

Apr 19, 2020

#1
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I graphed it and found that the two answers are  105  and  -105.

I haven't been able to figure out a way to do it algebraicly.

Apr 19, 2020
#2
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Hi Geno3141: Here is a solution courtesy of Mathematica 11 Home Edition:

Solve for x:
(x + 57)^(1/3) - (x - 57)^(1/3) = 6^(1/3)

Subtract (x + 57)^(1/3) from both sides:
-(x - 57)^(1/3) = 6^(1/3) - (x + 57)^(1/3)

Raise both sides to the power of three:
57 - x = (6^(1/3) - (x + 57)^(1/3))^3

Subtract (6^(1/3) - (x + 57)^(1/3))^3 from both sides:
57 - x - (6^(1/3) - (x + 57)^(1/3))^3 = 0

57 - x - (6^(1/3) - (x + 57)^(1/3))^3 = 108 + 3 6^(2/3) (x + 57)^(1/3) - 3 6^(1/3) (x + 57)^(2/3):
108 + 3 6^(2/3) (x + 57)^(1/3) - 3 6^(1/3) (x + 57)^(2/3) = 0

Simplify and substitute y = (x + 57)^(1/3).
108 + 3 6^(2/3) (x + 57)^(1/3) - 3 6^(1/3) (x + 57)^(2/3) = 108 + 3×6^(2/3) (x + 57)^(1/3) - 3 6^(1/3) ((x + 57)^(1/3))^2
= -3 6^(1/3) y^2 + 3 6^(2/3) y + 108:
-3 6^(1/3) y^2 + 3 6^(2/3) y + 108 = 0

Divide both sides by -3 6^(1/3):
y^2 - 6^(1/3) y - 6 6^(2/3) = 0

The left hand side factors into a product with three terms:
-(3 6^(1/3) - y) (y + 2 6^(1/3)) = 0

Multiply both sides by -1:
(3 6^(1/3) - y) (y + 2 6^(1/3)) = 0

Split into two equations:
3 6^(1/3) - y = 0 or y + 2 6^(1/3) = 0

Subtract 3 6^(1/3) from both sides:
-y = -3 6^(1/3) or y + 2 6^(1/3) = 0

Multiply both sides by -1:
y = 3 6^(1/3) or y + 2 6^(1/3) = 0

Substitute back for y = (x + 57)^(1/3):
(x + 57)^(1/3) = 3 6^(1/3) or y + 2 6^(1/3) = 0

Raise both sides to the power of three:
x + 57 = 162 or y + 2 6^(1/3) = 0

Subtract 57 from both sides:
x = 105 or y + 2 6^(1/3) = 0

Subtract 2 6^(1/3) from both sides:
x = 105 or y = -2 6^(1/3)

Substitute back for y = (x + 57)^(1/3):
x = 105 or (x + 57)^(1/3) = -2 6^(1/3)

Raise both sides to the power of three:
x = 105 or x + 57 = -48

Subtract 57 from both sides:
x = 105 or x = -105

(x + 57)^(1/3) - (x - 57)^(1/3) ⇒ (57 - 105)^(1/3) - (-57 - 105)^(1/3) = -(-6)^(1/3) ≈ -0.90856 - 1.57367 i
6^(1/3) ⇒ 6^(1/3) ≈ 1.81712:
So this solution is incorrect

(x + 57)^(1/3) - (x - 57)^(1/3) ⇒ (57 + 105)^(1/3) - (105 - 57)^(1/3) = 6^(1/3) ≈ 1.81712
6^(1/3) ⇒ 6^(1/3) ≈ 1.81712:
So this solution is correct

The solution is:

x = 105

Apr 19, 2020