Solve by Factoring x^(5/2)=32

Problem

x^(5/2)=32

Solution

1. Rewrite the equation to set it equal to zero by subtracting 32 from both sides.

x^(5/2)−32=0

2. Express the terms as powers with the same exponent to identify a factorable form.

(x^(1/2))^5−2^5=0

3. Apply the difference of powers formula a^n−b^n=(a−b)*(a^(n−1)+a^(n−2)*b+⋯+b^(n−1)) where a=x^(1/2) and b=2

(x^(1/2)−2)*(x^2+2*x^(3/2)+4*x+8*x^(1/2)+16)=0

4. Set the first factor equal to zero to find the real solution.

x^(1/2)−2=0

5. Isolate the variable by adding 2 to both sides.

x^(1/2)=2

6. Square both sides of the equation to solve for x

(x^(1/2))^2=2^2

x=4

Final Answer

x=4

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