Problem 36 - Elementary and 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 simplified result is \(5 - \sqrt{3}\).

Step 1: Applying the distributive property

Use the distributive property, also known as the FOIL method (First, Outside, Inside, Last), to expand the expression \((2 \sqrt{3} + 1)(\sqrt{3} - 1)\). Multiply each term in the first binomial by each term in the second binomial:- First: \(2 \sqrt{3} \times \sqrt{3} = 2(\sqrt{3})^2 = 2 \times 3 = 6\)- Outside: \(2 \sqrt{3} \times (-1) = -2 \sqrt{3}\)- Inside: \(1 \times \sqrt{3} = \sqrt{3}\)- Last: \(1 \times (-1) = -1\)

Step 2: Combine like terms

Now that each term has been multiplied out, combine like terms:Combine the constant terms: \(6 - 1 = 5\)Combine the terms with \(\sqrt{3}\):\(-2 \sqrt{3} + \sqrt{3} = -\sqrt{3}\) Thus, the expression simplifies to: \(5 - \sqrt{3}\).

Key Concepts

Distributive Property

The distributive property is like a special rule in math that helps us multiply expressions more easily. It tells us how to expand an expression that involves a binomial (which is two terms added together) multiplied by another binomial. You might have heard of this as the FOIL method, which stands for First, Outside, Inside, Last. This is a systematic way to ensure you don't miss any parts of the multiplication process. Let's see how it works:

• First: Multiply the first terms in each binomial. This is the product that sets the tone for the rest of the expansion.
• Outside: Multiply the outer terms in the binomial pair. These are the terms on the "outside" of your expression when written side by side.
• Inside: Multiply the inside terms, which sit right next to each other within the two binomials.
• Last: Finally, multiply the last terms of each binomial. These are the terms furthest to the right.

Understanding and applying the distributive property allows you to break down complex expressions into more manageable pieces that can be easily combined to simplify the expression.

Simplifying Expressions

After expanding an expression using the distributive property, you’ll often be left with several terms. The next step is to simplify the expression by combining like terms. Like terms are terms that have exactly the same variables raised to the same powers. In our example, once we applied FOIL, we got:

• Constant terms: These are plain numbers without variables. You need to add or subtract them after multiplying.
• Radical terms: Similar to variable terms, they can be combined if they have the same square root. For instance, \( -2 \sqrt{3} + \sqrt{3} = - \sqrt{3} \).

Simplification turns a complicated expression into something clearer and easier to work with. Knowing how to identify and combine like terms is essential in algebra and makes solving equations much more straightforward.

Square Roots

Square roots are a fundamental concept in mathematics where given a number, you are looking to find another number which, when multiplied by itself, gives the original number. For instance, the square of 3 is 9, thus the square root of 9 is 3, written as \(\sqrt{9} = 3 \).

In algebraic expressions, square roots can appear in various forms. When multiplying square roots, it’s important to remember that the square root of two numbers multiplied together is the same as the product of their square roots, like this: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab} \). Likewise, squaring a square root simply gives you the number inside: \( (\sqrt{n})^2 = n \).

When simplifying expressions that involve square roots, recognizing and manipulating these principles can help manage and simplify them, making complex computations much easier.