Solve for x x^4=16

Problem

x^4=16

Solution

1. Subtract 16 from both sides to set the equation to zero.

x^4−16=0

2. Recognize the expression as a difference of squares, where x^4=(x^2)^2 and 16=4^2

(x^2)^2−4^2=0

3. Factor using the difference of squares formula a^2−b^2=(a−b)*(a+b)

(x^2−4)*(x^2+4)=0

4. Factor the first term x^2−4 again using the difference of squares formula.

(x−2)*(x+2)*(x^2+4)=0

5. Set each factor equal to zero to find the real and imaginary roots.

x−2=0⇒x=2

x+2=0⇒x=−2

x^2+4=0⇒x^2=−4

6. Solve for x in the quadratic equation x^2=−4 by taking the square root of both sides.

x=±√(,−4)

x=±2*i

Final Answer

x=2,−2,2*i,−2*i

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