Solve by Factoring x^(2/3)-2x^(1/3)-8=0

Problem

x(2/3)−2*x(1/3)−8=0

Solution

1. Substitute a new variable to simplify the quadratic form by letting u=x(1/3)

2. Rewrite the equation in terms of u noting that u2=(x(1/3))2=x(2/3)

u2−2*u−8=0

3. Factor the quadratic expression by finding two numbers that multiply to −8 and add to −2

(u−4)*(u+2)=0

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

u−4=0⇒u=4

u+2=0⇒u=−2

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

x(1/3)=4

x(1/3)=−2

6. Cube both sides of each equation to isolate x

(x(1/3))3=4⇒x=64

(x(1/3))3=(−2)3⇒x=−8

Final Answer

x=64,−8

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