Solve for x x = sixth root of x

Problem

x=√(6,x)

Solution

1. Rewrite the radical expression using a fractional exponent.

x=x(1/6)

2. Raise both sides of the equation to the sixth power to eliminate the fractional exponent.

x6=(x(1/6))6

x6=x

3. Subtract x from both sides to set the equation to zero.

x6−x=0

4. Factor out the greatest common factor, which is x

x*(x5−1)=0

5. Set each factor equal to zero to find the potential solutions.

x=0

x5−1=0

6. Solve the second equation for x by adding 1 to both sides and taking the fifth root.

x5=1

x=√(5,1)

x=1

7. Verify the solutions in the original equation. For x=0 0=√(6,0) is true. For x=1 1=√(6,1) is true.

Final Answer

x={0,1}

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