Evaluate (-8+ square root of -12)/40

Problem

the fraction with numerator negative 8 plus the square root of negative 12 and denominator 40

Solution

1. Simplify the square root of the negative number by using the imaginary unit i where the square root of negative 1 equals i

the square root of negative 12 equals the square root of negative 1 times 4 times 3

the square root of negative 12 equals 2 i the square root of 3

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

the fraction with numerator negative 8 plus 2 i the square root of 3 and denominator 40

3. Factor out the greatest common factor from the terms in the numerator.

2 times open paren negative 4 plus i the square root of 3 close paren

4. Divide the numerator and the denominator by their greatest common factor, which is 2

2 times open paren negative 4 plus i the square root of 3 close paren over 40 equals the fraction with numerator negative 4 plus i the square root of 3 and denominator 20

5. Split the fraction into its real and imaginary parts to write it in standard form a plus b i

negative 4 over 20 plus i the square root of 3 over 20

6. Simplify the real part by reducing the fraction.

negative one fifth plus the fraction with numerator the square root of 3 and denominator 20 i

Final Answer

the fraction with numerator negative 8 plus the square root of negative 12 and denominator 40 equals negative one fifth plus the fraction with numerator the square root of 3 and denominator 20 i

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