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

Problem

x(3/2)=8

Solution

1. Subtract 8 from both sides to set the equation to zero.

x(3/2)−8=0

2. Rewrite the terms as cubes to prepare for the difference of cubes formula, noting that x(3/2)=(x(1/2))3 and 8=2

(x(1/2))3−2=0

3. Apply the difference of cubes formula, which is a3−b3=(a−b)*(a2+a*b+b2) where a=x(1/2) and b=2

(x(1/2)−2)*((x(1/2))2+2*x(1/2)+2)=0

4. Simplify the expression inside the parentheses.

(x(1/2)−2)*(x+2*x(1/2)+4)=0

5. Set the first factor to zero to find the real solution.

x(1/2)−2=0

6. Solve for x by isolating the radical and squaring both sides.

x(1/2)=2

(√(,x))2=2

x=4

7. Check the second factor for additional real solutions using the discriminant b2−4*a*c of the quadratic form u2+2*u+4 where u=x(1/2)

2−4*(1)*(4)=−12

Since the discriminant is negative, there are no further real solutions.

Final Answer

x=4

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