In Exercises 37–52, perform the indicated operations and write the result in standard form. __ ___ (3√−5 )( −4√−12 )

Verified step by step guidance

1. Recognize that the expression involves complex numbers because of the square roots of negative numbers.
2. Rewrite the square roots of negative numbers using imaginary units: \( \sqrt{-5} = \sqrt{5}i \) and \( \sqrt{-12} = \sqrt{12}i \).
3. Substitute these into the expression: \( (3\sqrt{5}i)(-4\sqrt{12}i) \).
4. Multiply the coefficients and the imaginary parts separately: \( 3 \times -4 = -12 \) and \( (\sqrt{5}i)(\sqrt{12}i) = \sqrt{60}i^2 \).
5. Recall that \( i^2 = -1 \), so the expression becomes \( -12 \times (-\sqrt{60}) \). Simplify \( \sqrt{60} \) and multiply to express the result in standard form \( a + bi \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations involving square roots of negative numbers, as seen in this problem.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. When multiplying two complex numbers, you must also remember that i² = -1, which can change the sign of the result. This concept is crucial for simplifying the expression given in the problem.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, the final result should be expressed in this form for clarity and consistency. This involves separating the real and imaginary parts after performing the necessary calculations.