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 problem involves multiplying two expressions with square roots of negative numbers. Recall that the square root of a negative number can be expressed using the imaginary unit i, where i = √-1.
2. Rewrite each square root in terms of i. For example, √-5 can be written as i√5, and √-12 can be written as i√12.
3. Substitute these rewritten forms into the expression: (3√-5)(-4√-12) becomes (3i√5)(-4i√12).
4. Simplify the multiplication step by step. Multiply the coefficients (3 and -4), the imaginary units (i and i), and the square root terms (√5 and √12). Remember that i * i = -1.
5. Combine the results into a single expression. Simplify the square root multiplication (√5 * √12 = √60) and reduce √60 to its simplest radical form if possible. Write the final result in standard form as a complex number (a + bi).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically 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 they allow us to express these roots in a meaningful way.

Multiplication of Complex Numbers

When multiplying complex numbers, the distributive property is applied, similar to multiplying polynomials. For example, when multiplying (a + bi)(c + di), the result is ac + adi + bci + bdi². Since i² = -1, this simplifies to (ac - bd) + (ad + bc)i. Mastery of this process is crucial for correctly performing operations on complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. This format is important because it clearly separates the real and imaginary components, making it easier to interpret and manipulate complex numbers. When performing operations, the final result should be presented in this standard form for clarity and consistency.