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))^5−3^5=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=3

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

x=9

Final Answer

x=9

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