Problem 43 - College Algebra Essentials

Question

In Exercises \(29-44\), perform the indicated operations and write the result in standard form. $$ (3 \sqrt{-5})(-4 \sqrt{-12}) $$

Step-by-Step Solution

Answer

The answer is \(0 - 24 \sqrt{15}i\)

Step 1: Multiply the coefficients

Multiplying \(3 * -4\) gives \( -12\). For the imaginary parts, we have \(\sqrt{-5} * \sqrt{-12}\), which gives us - \(\sqrt{60}\). So the expression becomes: \(-12 \sqrt{60} \)

Step 2: Convert the roots

To convert the roots to a standard form, take into account that \(\sqrt{-a}\)=i\(\sqrt{a}\), for a>0. Therefore, \(-12 \sqrt{60}\) becomes \(-12i \sqrt{60}\). Now, simplify \(\sqrt{60}\) to get \(2 \sqrt{15}\). Thus, the expression becomes \(-24i \sqrt{15}\).

Step 3: Write in standard form

The standard form of a complex number is \(a+bi\), thus the expression simplifies to \(0 - 24 \sqrt{15} i\).

Key Concepts

Standard Form of a Complex Number

Understanding the standard form of a complex number is essential in working with complex numbers. The standard form is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) representing the imaginary unit. This unit, \(i\), is defined by the property that \(i^2 = -1\).

In real-world terms, think of \(a\) as how far left or right you go on a number line, and \(bi\) as how far up or down you go. In the exercise given, we end up with the expression \(0 - 24 \sqrt{15} i\), which fits the standard form where the real part is 0 (hence, not shown) and the imaginary part is \(-24 \sqrt{15} i\). The real part doesn't always have to be visible, especially if it's zero, because adding zero doesn't change the value.

Multiplying Complex Numbers

To multiply complex numbers, follow the rules of algebraic multiplication, keeping in mind that \(i^2 = -1\). Start by multiplying the real numbers, then the imaginary numbers, and finally combine any like terms. In our case, the initial computation involved \(3 * -4\) for the real numbers, leading to \(-12\), and \(\sqrt{-5} * \sqrt{-12}\) for the imaginary numbers.

Multiplying the square roots of negative numbers introduces the imaginary unit \(i\) twice, which means you'll actually end up multiplying by \(-1\) too, as seen by converting each square root separately (for example, \(\sqrt{-5} = i\sqrt{5}\)). Ultimately, this results in the simplified form as a complex number. Always remember when dealing with the square root of a negative number, convert it to an imaginary form using \(i\), then simplify.

Imaginary Numbers

Introduction to Imaginary Numbers

Imaginary numbers are an expansion of the real number system where typically, square roots of negative numbers are not defined. The imaginary unit \(i\) enables these operations. Every imaginary number can be expressed as \(bi\) where \(b\) is a real number.

In our exercise example, the multiplication of square roots of negative numbers is simplified by using the imaginary unit. The \(\sqrt{-5}\) becomes \(i\sqrt{5}\), and similarly, \(\sqrt{-12}\) becomes \(i\sqrt{12}\). When you multiply these, you include \(i^2\), which simplifies to \(-1\) and turns our product into a real number times an imaginary unit. The power of imaginary numbers is in how they extend our capability to solve problems that would otherwise lack a solution in the real number system, such as taking the square root of a negative number.