Problem 36 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ (2 \sqrt{3}+1)(\sqrt{3}-1) $$

Step-by-Step Solution

Answer

The expression simplifies to \(5 - \sqrt{3}\).

Step 1: Identify the expression type

The expression \[ (2 \sqrt{3}+1)(\sqrt{3}-1) \] is in the form of \( (a+b)(c-d) \). We will use the distributive property, often called FOIL (First, Outside, Inside, Last), to expand it.

Step 2: Multiply the First terms

Multiply the first terms from each binomial: \( 2\sqrt{3} \times \sqrt{3} = 2 \times (\sqrt{3})^2 = 2 \times 3 = 6. \)

Step 3: Multiply the Outside terms

Multiply the outside terms: \( 2\sqrt{3} \times (-1) = -2\sqrt{3}. \)

Step 4: Multiply the Inside terms

Multiply the inside terms: \( 1 \times \sqrt{3} = \sqrt{3}. \)

Step 5: Multiply the Last terms

Multiply the last terms: \( 1 \times (-1) = -1. \)

Step 6: Combine all the terms

Combine all the products obtained in the previous steps: \[ 6 - 2\sqrt{3} + \sqrt{3} - 1. \]

Step 7: Simplify the expression

Combine like terms. Notice that \( -2\sqrt{3} \) and \( +\sqrt{3} \) are like terms, which combine to \( -\sqrt{3} \). Combine the constants \( 6 \) and \(-1\):\[ 6 - 1 = 5. \]Thus, the resulting expression is \[ 5 - \sqrt{3}. \]

Key Concepts

Distributive Property

The distributive property is an essential algebraic principle allowing us to simplify expressions and equations. It describes how multiplication interacts with addition and subtraction within parentheses. In essence, it lets us "distribute" a factor across terms inside the parentheses.

Consider the expression \[ (a+b)(c-d) = a(c-d) + b(c-d) \]Here, each term inside the first parentheses multiplies each term inside the second, effectively ensuring every possible combination of terms is covered. This principle is frequently applied using the acronym FOIL when dealing specifically with binomials.

If we were to distribute the terms properly:

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms in the sequence.
• Inside: Multiply the inner terms together.
• Last: Multiply the last terms in each group.

This method ensures that all interactions between terms are taken into account, leading to the correct expanded form of the expression.

Simplification of Expressions

Simplifying expressions is crucial for making complex expressions more manageable and easy to understand. When simplifying, the goal is to reduce the expression to its simplest form by combining like terms, performing arithmetic operations, and applying algebraic identities.

During simplification, always look for terms that can be combined. Like terms are terms that have identical variable parts. For instance, \(-2\sqrt{3}\) and \(+\sqrt{3}\) are like terms, because they both contain the square root of 3. These can be combined by summing coefficients, leading to \(-\sqrt{3}\) when combined.

Another aspect of simplification involves constants or numbers without variables. In everyday homework exercises, these might seem straightforward enough to combine. For example, reducing \(6 - 1\) gives \(5\). Each of these steps is vital in ensuring the expression is fully simplified and neatly reduced.

Binomials Multiplication

Multiplying binomials, such as \((2 \sqrt{3}+1)(\sqrt{3}-1)\), involves a systematic approach to ensure that every term in one binomial multiplies every term in the other. This comprehensive multiplication is why the distributive property and FOIL method are indispensable tools in this context.

When you apply the technique of multiplying binomials systematically:

• Start with the first terms, as with \(2\sqrt{3}\times\sqrt{3}\), yielding \(6\).
• Then, move to the outer terms, \(2\sqrt{3}\times -1\), resulting in \(-2\sqrt{3}\).
• Switch to the inner terms, \(1 \times \sqrt{3}\), giving \(\sqrt{3}\).
• Finally, manage the last terms, \(1 \times -1\), which is \(-1\).

Each pair-wise multiplication contributes to the expanded polynomial expression, ensuring all possible interactions between terms are evaluated and included. This technique not only ensures accuracy in expression evaluation but also enhances the understanding and fluency of dealing with algebraic expressions as a whole.