Problem 110 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ (5+i \sqrt{3})(2-i \sqrt{3}) $$

Step-by-Step Solution

Answer

The answer is \(13 - 3i \sqrt{3}\).

Step 1: Use the Distributive Property

To solve \[(5+i \sqrt{3})(2-i \sqrt{3})\], apply the distributive property (also known as the FOIL method for binomials):- Multiply the first terms: \(5 \times 2 = 10\)- Multiply the outer terms: \(5 \times (-i \sqrt{3}) = -5i \sqrt{3}\)- Multiply the inner terms: \(i \sqrt{3} \times 2 = 2i \sqrt{3}\)- Multiply the last terms: \(i \sqrt{3} \times (-i \sqrt{3}) = -(i^2)(3) = 3\) because \(i^2 = -1\).

Step 2: Combine Like Terms

We now combine the results of the multiplications from Step 1:- The real parts: \(10 + 3 = 13\)- The imaginary parts: \(-5i \sqrt{3} + 2i \sqrt{3} = -3i \sqrt{3}\)

Step 3: Write the Final Answer

Combine the results from Step 2 into the form \(a + bi\): The real part is \(13\) and the imaginary part is \(-3i \sqrt{3}\), so the expression simplifies to: \[13 - 3i \sqrt{3}\].

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra. It applies to expressions where a single term outside a parenthesis needs to be multiplied by each term inside the parenthesis. For example, when dealing with two binomials like \[(a + b)(c + d),\]we multiply each element in the first parenthesis with every element in the second:

• Multiply the first terms: \(a \times c\)
• Multiply the outer terms: \(a \times d\)
• Multiply the inner terms: \(b \times c\)
• Multiply the last terms: \(b \times d\)

This property extends to complex numbers, as they can also be treated with this method, simplifying complex expressions into an easier to manage form. Understanding this can be especially helpful when working on expressions involving variables, numbers, or complex numbers like in the original exercise.

FOIL Method

The FOIL method is a specific application of the distributive property used for multiplying binomials. FOIL stands for First, Outer, Inner, Last, and describes a systematic way to ensure all terms are multiplied correctly. To illustrate:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the final terms in each binomial.

For example, for the binomials \((5 + i \sqrt{3})(2 - i \sqrt{3})\), using the FOIL method ensures that no term is overlooked. This method parallels the distributive approach but gives a handy mnemonic for binomial multiplication.

Imaginary Unit

The imaginary unit, denoted as \(i\), is defined as the square root of negative one. This simple concept opens up an entirely new set of numbers called complex numbers. The key property of the imaginary unit is that \[i^2 = -1.\]This property is critical when multiplying complex numbers because whenever \(i^2\) appears in a multiplication, it can be replaced with \(-1\). This helps reduce expressions to a simpler form, seamlessly combining real and imaginary components.In the given exercise, recognizing that \(i^2 = -1\) allows us to simplify \(-(i^2)(3)\) to just \(3\), as was calculated when multiplying the last terms.

Binomials

A binomial is simply an algebraic expression containing two terms grouped with a plus or minus sign, presented as \((a + b)\) or \((a - b)\). The term 'binomial' derives from 'bi-' meaning two and 'nomial' meaning terms. These expressions frequently appear in algebra and can be expanded using methods like the distributive property or FOIL method. For instance, the original exercise features the binomials \((5 + i \sqrt{3})\) and \((2 - i \sqrt{3})\). These are often multiplied or added in many algebra problems, and understanding their structure is crucial. Each term in a binomial holds its unique place, whether it's the coefficient, variable, or complex component.