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How does \(2{x}^{5/3}+64=0\) have no solutions if the left side equals 0?

Guest Nov 28, 2017
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2x^(5/3) + 64 = 0

2x^(5/3) = -64    divide both sides by 2

x^(5/3) =-32        Take (5/3)th root of both sides

Simplify the following:
(-32)^(1/(5/3))

 

Multiply the numerator of 1/(5/3) by the reciprocal of the denominator. 1/(5/3) = (1×3)/5:
(-32)^(3/5)

 

(-32)^(3/5) = ((-32)^3)^(1/5) = ((-1)^3×32^3)^(1/5) = ((-1)^3×2^15)^(1/5):
((-1)^3 2^15)^(1/5)

 

((-1)^3 2^15)^(1/5) = ((-1)^3)^(1/5) (2^15)^(1/5) = (-1)^(3/5)×2^(15/5) = (-1)^(3/5)×8:


x = 8×(-1)^(3/5)

 

Proof: 2 * [ 8*(-1)^(3/5)]^(5/3) =-64 + 64 =0

Guest Nov 28, 2017
edited by Guest  Nov 28, 2017

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