Solve for x x^(4/3)-5x^(2/3)+6=0

Problem

x(4/3)−5*x(2/3)+6=0

Solution

1. Identify the equation as being in quadratic form by noticing that x(4/3) is the square of x(2/3)

2. Substitute a new variable u to simplify the equation, letting u=x(2/3)

u2−5*u+6=0

3. Factor the quadratic equation into two binomials.

(u−2)*(u−3)=0

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

u=2

u=3

5. Back-substitute x(2/3) for u to find the values of x

x(2/3)=2

x(2/3)=3

6. Isolate x by raising both sides of each equation to the power of 3/2

x=2(3/2)

x=3(3/2)

7. Simplify the radical expressions.

x=√(,2)=√(,8)=2√(,2)

x=√(,3)=√(,27)=3√(,3)

Final Answer

x=2√(,2),3√(,3)

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