Solve by Factoring x^(5/2)=243

Problem

x(5/2)=243

Solution

1. Rewrite the equation to set it equal to zero by subtracting 243 from both sides.

x(5/2)−243=0

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

(x(1/2))5−3=0

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

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

4. Set the first factor to zero to find the real solution, as the second factor has no real roots for x(1/2)

x(1/2)−3=0

5. Isolate the variable by adding 3 to both sides.

x(1/2)=3

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

(x(1/2))2=3

x=9

Final Answer

x=9

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