Problem 77 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$(3 \sqrt{2}+\sqrt{3})(2 \sqrt{3}-\sqrt{2})$$

Step-by-Step Solution

Answer

The simplified expression is \(5 \sqrt{6}\).

Step 1: Recognize the Expression Type

The expression provided is a product of two binomials: \((3 \sqrt{2} + \sqrt{3})(2 \sqrt{3} - \sqrt{2})\). To simplify this, we'll use the distributive property (or FOIL method for binomials).

Step 2: Apply Distributive Property

We will expand the expression by multiplying each term in the first binomial by each term in the second binomial:1. \(3 \sqrt{2} \times 2 \sqrt{3}\)2. \(3 \sqrt{2} \times (-\sqrt{2})\)3. \(\sqrt{3} \times 2 \sqrt{3}\)4. \(\sqrt{3} \times (-\sqrt{2})\)

Step 3: Perform the Multiplication

Calculate each product:1. \(3 \sqrt{2} \times 2 \sqrt{3} = 6 \sqrt{6}\)2. \(3 \sqrt{2} \times (-\sqrt{2}) = -3 \times 2 = -6\)3. \(\sqrt{3} \times 2 \sqrt{3} = 2 \times 3 = 6\)4. \(\sqrt{3} \times (-\sqrt{2}) = -\sqrt{6}\)

Step 4: Combine Like Terms

Combine the terms obtained:- \(6 \sqrt{6} - \sqrt{6} = 5 \sqrt{6}\)- Combine the constant terms: \(-6 + 6 = 0\)So, the expression simplifies to \(5 \sqrt{6}\).

Key Concepts

Binomials

Let's dive into the world of binomials! A binomial is a type of algebraic expression that consists of two terms. For example, in the expression \[(3 \sqrt{2} + \sqrt{3})(2 \sqrt{3} - \sqrt{2})\], we have two binomials:

• \(3 \sqrt{2} + \sqrt{3}\)
• \(2 \sqrt{3} - \sqrt{2}\)

A good way to think about binomials is like having two baskets, each containing terms that we will eventually need to combine or multiply. Understanding and recognizing binomials is crucial because it guides us on how to apply mathematical operations like multiplication effectively.

When multiplying binomials, a common method used is the FOIL method, which stands for First, Outside, Inside, and Last. This method ensures that each term in the first binomial multiplies with each term in the second. This comprehensive approach helps in thoroughly expanding the expression before simplifying it. Think of it buying groceries by itemizing each basket separately and ensuring nothing is left out.

Distributive Property

The distributive property is a powerful tool in algebra, especially when dealing with expressions like the one we're simplifying. Essentially, this property allows us to multiply a single term by every term within a parenthesis. For our exercise, we apply it to multiply two binomials.

Here's how it works:

• We take the first term from the first binomial: \(3 \sqrt{2}\), and multiply it by both terms of the second binomial, \(2 \sqrt{3} - \sqrt{2}\).
• Then, we take the second term of the first binomial: \(\sqrt{3}\), and do the same, multiplying it by each term of the second binomial.

By carrying out these multiplications separately and then adding the results, we effectively distribute each term of one binomial across the other, ensuring that all possible products between terms are covered. Remember, distributing in this context doesn't refer to parceling out items like in logistics, but rather spreading multiplication evenly across terms.

Real Numbers

Real numbers form the backbone of most algebraic operations. They include all numbers that you can find on the number line, such as positive and negative integers, fractions, and irrational numbers like \(\sqrt{2}\) and \(\sqrt{3}\).

In this exercise, all variables are assumed to represent nonnegative real numbers. This assumption means we won't encounter undefined expressions or the need to deal with imaginary numbers.

Understanding the nature of real numbers allows us to apply the operations comfortably, such as multiplication of radicals. For instance, when we see terms like \(\sqrt{2} \times \sqrt{3}\), they combine using the property of radicals, simplifying to \(\sqrt{6}\).

These properties ensure the operations remain defined and real, making our simplifications both valid and reliable. Think of real numbers as the canvases of algebra; they provide the necessary backdrop for all kinds of mathematical painting and operation.