Solve x^1/3=2+15x^-1/3

Question

Any help would be so appreciated

Answer 1

Lauren's answer is correct. I will add a few more steps to this.

Convert that negative exponent to positive by moving it to the denominator

x^1/3=2+15/x^1/3

Multiply both sides of the equation by x^1/3 to simplify and get rid of the fraction. I did create the fraction, but thought it would be easier to follow with this extra step.

(x^1/3=2+15/x^1/3)x^1/3

Distribute x^1/3

x^2/3=2x^1/3+15 (For the first term, the exponents add together) Move everything to one side of the equation

x^2/3-2x^1/3-15 =2x^1/3+15 2x^1/3-15 Simplify

x^2/3-2x^1/3-15 = 0

The easiest way to solve from here is to factor. The easiest way to factor is by u substitution. Let u=x^1/3

The equation becomes: u²-2u-15=0

u²-2u-15=0

Factor

(u-5)(u+3)=0

Solve both equations for u

u-5=0 Add 5 to both sides

u=5

OR

u+3=0 Subtract 3 from both sides

u=-3

Unfortunately, we have u but we need x! Almost there.

Plug u into this to solve for x: u=x^1/3

5=x^1/3 Cube both sides to solve for x

5³=(x^1/3)³

125=x

-3=x^1/3 Cube both sides to solve for x

(-3)³=(x^1/3)³

-27=x

Answer 2

0 = -x^1/3 + 2 + 15x^-1/3

0 = (-x^2/3 + 2x^1/3 + 15) / x^1/3 (multiply by 1 as x^1/3 / x^1/3)

0 = -(x^1/3 - 5)(x^1/3 + 3) / x^1/3

x^1/3 = 5, x = 125

x^1/3 = -3, x = -27