Evaluate (-20+ square root of -75)/40

Problem

(−20+√(,−75))/40

Solution

1. Identify the square root of a negative number as an imaginary number using the property √(,−a)=i√(,a)

√(,−75)=i√(,75)

2. Simplify the radical by factoring out the largest perfect square, which is 25

√(,75)=√(,25⋅3)=5√(,3)

3. Substitute the simplified radical back into the original expression.

(−20+5*i√(,3))/40

4. Factor out the greatest common factor from the numerator, which is 5

5*(−4+i√(,3))

5. Divide the numerator and denominator by 5 to simplify the fraction.

(5*(−4+i√(,3)))/40=(−4+i√(,3))/8

6. Rewrite the expression in standard complex form a+b*i by splitting the fraction.

(−4)/8+(i√(,3))/8

7. Reduce the real part to its simplest form.

−1/2+√(,3)/8*i

Final Answer

(−20+√(,−75))/40=−1/2+√(,3)/8*i

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