Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (2-\sqrt{-4})(3-\sqrt{-16}) $$

Step-by-Step Solution

Answer

The expression is \(-2 - 14i\).

Step 1: Identify Imaginary Numbers

The expression contains square roots of negative numbers: \( \sqrt{-4} \) and \( \sqrt{-16} \). Recognize these as imaginary numbers. Specifically, \( \sqrt{-4} = 2i \) and \( \sqrt{-16} = 4i \) since \( \sqrt{-1} = i \).

Step 2: Substitute Imaginary Numbers

Substitute the square roots of negative numbers in the original expression:\((2 - \sqrt{-4})(3 - \sqrt{-16}) = (2 - 2i)(3 - 4i)\).

Step 3: Apply the Distributive Property

Use the distributive property (FOIL method) to expand the expression \((2 - 2i)(3 - 4i)\):\(2(3) + 2(-4i) - 2i(3) + (-2i)(-4i)\).

Step 4: Calculate Each Term

Calculate each component of the expanded expression:1. \(2(3) = 6\)2. \(2(-4i) = -8i\)3. \(-2i(3) = -6i\)4. \(-2i(-4i) = 8i^2\).

Step 5: Simplify using \(i^2 = -1\)

Substitute \(i^2 = -1\) into the equation. This turns the imaginary square term:\(8i^2 = 8(-1) = -8\).

Step 6: Combine Real and Imaginary Parts

Combine all real and imaginary components from the last step:For real parts: \(6 + (-8) = -2\) For imaginary parts: \(-8i - 6i = -14i\).{

Step 7: Write the Final Expression

Combine the results into the form \(a + bi\):\(-2 - 14i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are quite fascinating! They first appear when you try to take the square root of a negative number. Normally, in the realm of real numbers, you can't find the square root of a negative number. This is where imaginary numbers, denoted by \(i\), come into play. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).

When you take the square root of a negative number, you use \(i\) to express the result. For example, \(\sqrt{-4}\) becomes \(2i\). Why? Because \(\sqrt{-4} = \sqrt{4 \cdot -1}\), which simplifies to \(\sqrt{4} \cdot \sqrt{-1} = 2 \cdot i = 2i\). Breaking the expression into these parts helps to understand and manage calculations with complex numbers.

Some key points about imaginary numbers are:

• \(i^2 = -1\)
• Fractions involving \(i\) can also allow calculations with complex numbers.
• They are not visible on the usual number line, but crucial in advanced mathematics, physics, and engineering.

Distributive Property

The distributive property is a fundamental mathematical concept that comes in very handy when expanding expressions. In the context of the problem at hand, the expression \((2 - 2i)(3 - 4i)\) is expanded using this property. The distributive property allows you to multiply each term inside a set of parentheses by each term in another set.

Here's how it works:

• Multiply the first terms: \(2 \cdot 3\)
• Multiply the outer terms: \(2 \cdot -4i\)
• Multiply the inner terms: \(-2i \cdot 3\)
• Multiply the last terms: \(-2i \cdot -4i\)

By applying this property, you ensure that every term in the first binomial is multiplied by every term in the second. Expanding expressions this way is crucial for reaching the simplified form, especially when you're dealing with complex numbers.

FOIL Method

The FOIL method is a straightforward technique used to multiply two binomials. This acronym stands for First, Outer, Inner, Last, which describes the sequence in which you multiply the terms:

• First: Multiply the first terms of each binomial. For instance, in \((2 - 2i)(3 - 4i)\), you multiply \(2\) and \(3\) to get \(6\).
• Outer: Multiply the outer terms. This means \(2 \) and \(-4i\), resulting in \(-8i\).
• Inner: Multiply the inner terms, \(-2i\) and \(3\), which equals \(-6i\).
• Last: Multiply the last terms of the binomials. In this example, \(-2i\) and \(-4i\) yield \(8i^2\).

The FOIL method offers a structured way to ensure you don't miss any term when expanding two binomials. It's particularly useful when handling expressions involving complex numbers and can simplify the computation process significantly.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. This involves combining like terms and applying known mathematical identities. In the context of complex numbers, one helpful identity is \(i^2 = -1\).

After applying the FOIL method to \((2 - 2i)(3 - 4i)\), you get \(6 - 8i - 6i + 8i^2\). To simplify this expression, you need to combine like terms and substitute \(i^2\) with \(-1\). So, \(8i^2\) becomes \(8(-1) = -8\).

Now, arrange the expression using the simplified terms:

• Combine the real numbers: \(6 + (-8) = -2 \)
• Combine the imaginary numbers: \(-8i - 6i = -14i\)

So the simplified form is \(-2 - 14i\), written neatly as a complex number, with a real part \(a = -2\) and an imaginary part \(b = -14\).

This final step in algebra not only tidies up the expression but also makes the result much easier to interpret.