Evaluate fourth root of -625

Problem

√(4,−625)

Solution

1. Identify the nature of the expression. The task asks for the fourth root of a negative number, which is not a real number because any real number raised to an even power is non-negative.

2. Express the number using the imaginary unit i where i2=−1 We can write the expression as the fourth root of the product of 625 and −1

3. Determine the principal complex root using the formula for roots of complex numbers. A complex number z=r*(cos(θ)+i*sin(θ)) has nth roots given by (w_k)=√(n,r)*(cos((θ+2*k*π)/n)+i*sin((θ+2*k*π)/n))

4. Substitute the values for −625 Here, r=625 and θ=π The principal root (k=0 is found by taking the fourth root of the magnitude and using the angle π/4

5. Calculate the magnitude.

√(4,625)=5

6. Calculate the trigonometric components for the principal root.

cos(π/4)=√(,2)/2

sin(π/4)=√(,2)/2

7. Combine the results to find the principal value.

5*(√(,2)/2+i√(,2)/2)=(5√(,2))/2+(5√(,2))/2*i

Final Answer

√(4,−625)=(5√(,2))/2+(5√(,2))/2*i

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