Problem 20 - Algebra 2

Question

Multiply each pair of conjugates. $$ (4-2 \sqrt{3})(4+2 \sqrt{3}) $$

Step-by-Step Solution

Answer

The result of the multiplication is \(4\).

Step 1: Identify the Terms

In this case \(a=4\) and \(b=2\sqrt{3}\). The expressions \(4-2\sqrt{3}\) and \(4+2\sqrt{3}\) are conjugate pairs. Conjugate pairs have the form \(a+b\) and \(a-b\).

Step 2: Apply the Conjugate Pairs Rule

The conjugate pairs rule is \((a+b)(a-b)=a^2-b^2\). Applying this rule gives \((4)^2-(2\sqrt{3})^2\).

Step 3: Perform the Calculations

Square each number, yielding \(16-4*3\). Then, simplify \(16-12\) to get \(4\).

Key Concepts

Conjugate Pairs Rule

When working with conjugate pairs, it's important to recognize their structure. Conjugate pairs are two expressions of the form \(a + b\) and \(a - b\). These pairs play a crucial role in mathematics, especially when it involves complex numbers and radicals.
Conjugate pairs are multiplied using the conjugate pairs rule, which states that

• \((a+b)(a-b) = a^2 - b^2\)

This formula allows us to bypass the longer process of distributing each term. For example, in our exercise, we identify \(a = 4\) and \(b = 2\sqrt{3}\).
After recognizing the terms, the conjugate pairs rule simplifies the multiplication to

• \((4)^2 - (2\sqrt{3})^2\)

By squaring the individual terms, the expression becomes much simpler without having to multiply every term with each other.

Multiplication of Radicals

Radicals are numbers that involve a root, most commonly the square root. Multiplying such numbers can initially seem complex, but it follows straightforward rules.

• When multiplying radicals, we use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

In our specific example, \(b = 2\sqrt{3}\), which involves multiplying the radical by itself. When you square a radical, the square and the square root cancel out:
\((2\sqrt{3})^2 = 4 \cdot 3 = 12\).
This simplifies the process greatly, as we no longer have to deal with the radicals after squaring.

Simplification of Expressions

Simplification in mathematics refers to making an expression as compact and simple as possible. After identifying and using key rules such as those for conjugates and radicals, the final step in any algebraic problem is usually simplification.
In our example where the expression simplifies to \(16 - 12\), the last task to make is:

• Subtract to get \(4\).

This step reduces the expression to a single number or simplest form, which is easy to interpret and understand. The ability to simplify is essential in algebra as it enables you to solve equations efficiently and with minimal errors.
Recognize the initial forms and operations and perform them step by step to achieve a clean, concise solution.