Evaluate fourth root of -1000

Problem

√(4,−1000)

Solution

1. Identify the expression as the fourth root of a negative number, which involves complex numbers since the index is even.

2. Rewrite the expression using the imaginary unit i by noting that √(4,−1000)=√(4,1000)⋅√(4,−1)

3. Simplify the real part √(4,1000) by factoring out perfect powers: √(4,10)=10(3/4)

4. Apply Euler's formula to find the principal root of √(4,−1) which is e(i*π/4)=cos(π/4)+i*sin(π/4)=√(,2)/2+√(,2)/2*i

5. Combine the terms to find the principal value: 10(3/4)*(√(,2)/2+√(,2)/2*i)

6. Distribute the constant to reach the final form.

Final Answer

√(4,−1000)=(√(4,1000)√(,2))/2+(√(4,1000)√(,2))/2*i

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