Solve for x x^4=81

Problem

x^4=81

Solution

1. Rewrite the equation by moving all terms to one side to set the equation to zero.

x^4−81=0

2. Recognize the expression as a difference of squares, where x^4=(x^2)^2 and 81=9^2

(x^2−9)*(x^2+9)=0

3. Factor the first term further using the difference of squares identity again, since x^2−9=x^2−3^2

(x−3)*(x+3)*(x^2+9)=0

4. Apply the zero product property by setting each factor equal to zero to find all possible solutions.

x−3=0⇒x=3

x+3=0⇒x=−3

x^2+9=0⇒x^2=−9

5. Solve for the remaining roots by taking the square root of both sides of the quadratic equation involving the imaginary unit i

x=±√(,−9)

x=±3*i

Final Answer

x=3,−3,3*i,−3*i

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