Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-3+\sqrt{-25})(8-\sqrt{-36}) $$

Step-by-Step Solution

Answer

The expression is simplified to \(6 + 58i\).

Step 1: Simplify Square Roots of Negative Numbers

First, we simplify the square roots of negative numbers using the imaginary unit, which is defined as \(i = \sqrt{-1}\). Therefore, \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i\) and \(\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i\). Now the expression becomes \((-3 + 5i)(8 - 6i)\).

Step 2: Distribute the Terms

To multiply the expression \((-3 + 5i)(8 - 6i)\), we use the distributive property (FOIL method): - Multiply \(-3\) by \(8\): \(-3 \cdot 8 = -24\).- Multiply \(-3\) by \(-6i\): \(-3 \cdot -6i = 18i\).- Multiply \(5i\) by \(8\): \(5i \cdot 8 = 40i\).- Multiply \(5i\) by \(-6i\): \(5i \cdot -6i = -30i^2\).

Step 3: Simplify the Terms

Now, add these results together: \(-24 + 18i + 40i - 30i^2\). The term \(-30i^2\) simplifies to \(30\) because \(i^2 = -1\). Thus, the expression becomes:\(-24 + 58i + 30\).

Step 4: Combine Real Parts and Imaginary Parts

Combine the real parts and the imaginary parts separately: - Real part: \(-24 + 30 = 6\).- Imaginary part: \(58i\).So, the expression is simplified to \(6 + 58i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of \(-1\), making it unique and essential when dealing with numbers that involve square roots of negative values. In mathematics, the symbol \(i\) is specifically used to handle equations where a negative value appears under a square root.

When you encounter a square root of a negative number, such as \(\sqrt{-25}\), it can be simplified using the imaginary unit. This is done by separating the negative sign as \(\sqrt{-1} = i\), allowing for further calculations. For instance:

• \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i\)
• \(\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i\)

With these transformations, problems involving negative square roots become manageable, and you can proceed with operations as you would with real numbers, keeping \(i\) in its place.

Distributive Property

The distributive property is a key principle used when multiplying expressions, particularly those involving complex numbers. It allows us to systematically expand an expression by distributing each term across another. In this exercise, we use the distributive property to simplify the product \((-3 + 5i)(8 - 6i)\).

This method, often associated with the acronym FOIL (First, Outer, Inner, Last), involves multiplying each pair of terms from the binomials:

• First: \(-3 \times 8 = -24\)
• Outer: \(-3 \times -6i = 18i\)
• Inner: \(5i \times 8 = 40i\)
• Last: \(5i \times -6i = -30i^2\)

After performing these multiplications, you sum up the results. Note the importance of recognizing \(i^2 = -1\), which significantly transforms \(-30i^2\) into \(30\). This step-by-step process simplifies complex expressions and is crucial when working with algebraic manipulations.

Simplifying Expressions

Simplifying expressions involves combining like terms and arranging the expression into a more readable and compact form. With complex numbers, this means separately dealing with real and imaginary components, then combining them whenever possible.

Consider the expression: \(-24 + 18i + 40i - 30i^2\). Here, you:

• Recognize and simplify \(-30i^2\) to \(30\) using \(i^2 = -1\)
• Combine imaginary terms: \(18i + 40i = 58i\)
• Add real numbers: \(-24 + 30 = 6\)

Thus, the expression ultimately simplifies to \(6 + 58i\), where 6 is the real part, and 58i is the imaginary part. This organized approach ensures clarity and accuracy in results, making complex numbers more intuitive to work with.