Evaluate (-12+ square root of -45)/24

Problem

(−12+√(,−45))/24

Solution

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

√(,−45)=i√(,45)

2. Simplify the radical by factoring out the largest perfect square, which is 9

√(,45)=√(,9⋅5)=3√(,5)

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

(−12+3*i√(,5))/24

4. Factor out the greatest common factor from the numerator, which is 3

3*(−4+i√(,5))

5. Divide both the numerator and the denominator by 3 to simplify the fraction.

(3*(−4+i√(,5)))/24=(−4+i√(,5))/8

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

(−4)/8+(i√(,5))/8

7. Reduce the real part to its simplest form.

−1/2+√(,5)/8*i

Final Answer

(−12+√(,−45))/24=−1/2+√(,5)/8*i

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