Evaluate fourth root of -256

Problem

√(4,−256)

Solution

1. Identify the expression as the fourth root of a negative number. In the set of real numbers, the even root of a negative number is undefined. However, we can evaluate this in the complex number system.

2. Rewrite the expression using the imaginary unit i where i2=−1

√(4,−256)=√(4,256⋅(−1))

3. Apply the property of roots √(n,a*b)=√(n,a)√(n,b)

√(4,256)⋅√(4,−1)

4. Calculate the fourth root of the positive constant. Since 4=256

√(4,256)=4

5. Determine the fourth roots of −1 using Euler's formula e(i*θ)=cos(θ)+i*sin(θ) The number −1 can be written as e(i*(π+2*k*π)) The roots are given by e(i(π+2*k*π)/4) for k=0,1,2,3

k=0⇒e(iπ/4)=cos(π/4)+i*sin(π/4)=√(,2)/2+√(,2)/2*i

k=1⇒e(i(3*π)/4)=cos((3*π)/4)+i*sin((3*π)/4)=−√(,2)/2+√(,2)/2*i

k=2⇒e(i(5*π)/4)=cos((5*π)/4)+i*sin((5*π)/4)=−√(,2)/2−√(,2)/2*i

k=3⇒e(i(7*π)/4)=cos((7*π)/4)+i*sin((7*π)/4)=√(,2)/2−√(,2)/2*i

6. Multiply the constant 4 by each root of −1 to find all four complex roots.

4⋅(√(,2)/2±√(,2)/2*i)=2√(,2)±2√(,2)*i

4⋅(−√(,2)/2±√(,2)/2*i)=−2√(,2)±2√(,2)*i

Final Answer

√(4,−256)=±2√(,2)±2√(,2)*i

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