Solve by Factoring x^(3/2)=64

Problem

x(3/2)=64

Solution

1. Set the equation to zero by subtracting 64 from both sides to prepare for factoring.

x(3/2)−64=0

2. Rewrite the terms as cubes to identify a difference of cubes pattern, noting that x(3/2)=(x(1/2))3 and 64=4

(x(1/2))3−4=0

3. Apply the difference of cubes formula, which states a3−b3=(a−b)*(a2+a*b+b2) where a=x(1/2) and b=4

(x(1/2)−4)*((x(1/2))2+4*x(1/2)+4)=0

4. Simplify the factors inside the expression.

(x(1/2)−4)*(x+4*x(1/2)+16)=0

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

x(1/2)−4=0

6. Isolate the variable by adding 4 to both sides.

x(1/2)=4

7. Square both sides to solve for x

(x(1/2))2=4

x=16

8. Evaluate the second factor x+4*x(1/2)+16=0 using the quadratic formula for x(1/2) This factor yields no real solutions because the discriminant 4−4*(1)*(16)=−48 is negative.

Final Answer

x=16

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