Solve by Factoring x^(2/3)-x^(1/3)-6=0

Problem

x(2/3)−x(1/3)−6=0

Solution

1. Substitute a new variable to simplify the equation. Let u=x(1/3)

2. Rewrite the equation in terms of u Since u2=(x(1/3))2=x(2/3) the equation becomes:

u2−u−6=0

3. Factor the quadratic expression by finding two numbers that multiply to −6 and add to −1 These numbers are −3 and 2

(u−3)*(u+2)=0

4. Solve for u by setting each factor to zero:

u−3=0⇒u=3

u+2=0⇒u=−2

5. Back-substitute x(1/3) for u to find the values of x

x(1/3)=3

x(1/3)=−2

6. Cube both sides of each equation to isolate x

(x(1/3))3=3⇒x=27

(x(1/3))3=(−2)3⇒x=−8

Final Answer

x=27,−8

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