Problem 44 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(3 \sqrt{-7})(2 \sqrt{-8})$$

Step-by-Step Solution

Answer

The result of the operation is \( -12 \sqrt{14} \)

Step 1: Represent Negative Square Roots as Complex Numbers

Represent each negative square root as a complex number using \(i\). Thus, we have \(3 \sqrt{-7}\) can be written as \(3 \sqrt{7} i\), and \(2 \sqrt{-8}\) can be written as \(2 \sqrt{8} i\).

Step 2: Simplify the Square Roots

Simplify the square roots \( \sqrt{7} \) and \( \sqrt{8} \). Since 7 is a prime number, it cannot be simplified further. However, 8 can be simplified as \(2 \sqrt{2}\). This gives us the simplified forms of the complex numbers as \(3 \sqrt{7}i\) and \(4 \sqrt{2}i\).

Step3: Multiply the Complex Numbers

Now, multiply the numbers: \( (3 \sqrt{7}i) \times (4 \sqrt{2}i) \). First, multiply the real parts together to get \(12 \sqrt{14}\). Second, multiply the imaginary parts together to get \(i \times i = -1\). Therefore, the result is \( -12 \sqrt{14} \).

Key Concepts

Negative Square Roots

Square roots can sometimes involve negative numbers, and this can be confusing at first. In standard arithmetic, you can't take a square root of a negative number because no real number multiplied by itself will give a negative number. This is where complex numbers come in.

To handle negative square roots, we use the imaginary unit, denoted by the letter 'i'.

• For any negative number, its square root is expressed using 'i'.
• Thus, the square root of -1 is 'i', and the square root of -x is \( \sqrt{x} i \).

In our example, \( \sqrt{-7} \) is expressed as \( \sqrt{7} i \), and \( \sqrt{-8} \) becomes \( \sqrt{8} i \). By expressing negative square roots this way, we can perform further operations more easily.

Standard Form

When working with complex numbers, it's often necessary to write them in a 'standard form'. This standard form is a simple way of stating the number consisting of both a real part and an imaginary part. The standard form of a complex number is \( a + bi \) where:

• 'a' is the real part
• 'b' is the imaginary part
• 'i' is the imaginary unit

This form helps in standardizing the operations between complex numbers. In the given exercise, standard form becomes important when simplifying and representing the final product, even though the real part might be zero, making it purely imaginary like \( -12 \sqrt{14} \).

Imaginary Unit

The imaginary unit, represented as 'i', is a fundamental part of complex numbers. It represents the square root of -1. Once you use 'i', standard operations of algebra can be extended to include complex numbers.

• 'i' is defined such that \( i^2 = -1 \)
• Using 'i' allows us to express numbers that would otherwise be difficult or impossible to return as real numbers alone

In calculations involving complex numbers, 'i' helps to translate results obtained from operations involving negative square roots, providing a foundation for complex arithmetic.

Complex Multiplication

Complex multiplication sounds challenging but follows a simple rule: distribute just like when multiplying algebraic expressions. For two complex numbers \( (a + bi) \) and \( (c + di) \), multiply as follows:

• Multiply the real parts \( ac \)
• Multiply the real by complex parts \((ad)i + (bc)i \)
• Multiply the complex parts \((bd)i^2\)
• Remembering that \(i^2 = -1\), combine terms to get another standard form result

In our exercise, after simplifying the negative square roots, the multiplication was carried out with similar steps to achieve the result \( -12 \sqrt{14} \), showing this in simple terms even improves calculation efficiency.