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solve 2x^2/3-5x^1/3=3

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2 Answers

The equation is 2x2/3 - 5x/3 = 3. Let's make the coefficients integers to make life easier by multiplying through by 3. Then we move the right hand side term to the left.

2x2/3 - 5x/3 = 3

2x2 - 5x = 9

2x2 - 5x - 9 = 0

Now use quadratic formula since it doesn't factor:

x = (-(-5) ± √((-5)2-4*2*(-9))) / (2*2) = (5 ± √97) / 4

Or else 2x^(2/3) - 5x^(1/3)=3

Divide by x^(1/3), 2x^(1/3) - 5 = 3x(-1/3)

Divide by 2 and rearrange, x^(1/3) - 5/2 - (3/2)x^(-1/3)=0

This will factor to (x^(1/6) +(1/2)x^(-1/6)) (x^(1/6) - 3x^(-1/6))=0

x^(1/6) = -(1/2)x^(-1/6)

x^(1/3) = -1/2

x= -1/8

Likewise the other term solves to x=27

(If this is the correct solution, you really need to include parenthesis in your exponents.  It eliminates confusion and will serve you well in college.  Profs have no time or patience for ambiguity.