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