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 multiplying two complex numbers in the form of \( a\sqrt{-1} \), which can be rewritten using the imaginary unit \( i = \sqrt{-1} \). So, rewrite \( \sqrt{-5} \) as \( \sqrt{5}i \) and \( \sqrt{-12} \) as \( \sqrt{12}i \).
2. Rewrite the original expression \( (3\sqrt{-5})(-4\sqrt{-12}) \) as \( (3\sqrt{5}i)(-4\sqrt{12}i) \).
3. Multiply the coefficients and the square root parts separately: multiply \( 3 \times -4 \) and \( \sqrt{5} \times \sqrt{12} \), and multiply the imaginary units \( i \times i \).
4. Recall that \( i \times i = i^2 = -1 \), so replace \( i^2 \) with \( -1 \) in your expression.
5. Simplify the expression by multiplying all numerical parts and applying the \( -1 \) from \( i^2 \), then write the final result in the form \( a + bi \), which is the standard form for complex numbers.

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form requires separating and simplifying these parts after performing operations like multiplication.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL) and applying i² = -1 to simplify terms. This process combines real and imaginary parts to form a new complex number.

Simplifying Square Roots of Negative Numbers

Square roots of negative numbers are expressed using imaginary unit i, where √(-n) = i√n. Simplifying these roots is essential before performing multiplication to correctly handle the imaginary components.