Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (\sqrt{6}+6)(\sqrt{6}-7) $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(-36 - \sqrt{6}\).

Step 1: Identify the Formula

We recognize this as a multiplication of two binomials, specifically a product of the form \((a+b)(a+c)\). In this case, \(a = \sqrt{6}\), \(b = 6\), and \(c = -7\). We will apply this formula to expand the expression.

Step 2: Apply Distribution

Using the distributive property, we expand the expression\((\sqrt{6} + 6)(\sqrt{6} - 7)\) as follows:- First, multiply the first terms: \(\sqrt{6} \times \sqrt{6} = 6\).- Then, the outer terms: \(\sqrt{6} \times -7 = -7\sqrt{6}\).- Next, the inner terms: \(6 \times \sqrt{6} = 6\sqrt{6}\).- Finally, the last terms: \(6 \times -7 = -42\).

Step 3: Combine Like Terms

Now that we have distributed the terms, we combine them to form a single expression:\[ 6 - 7\sqrt{6} + 6\sqrt{6} - 42 \].Notice that the \(-7\sqrt{6}\) and \(6\sqrt{6}\) are like terms, so we combine them:- \(-7\sqrt{6} + 6\sqrt{6} = -\sqrt{6}\).

Step 4: Simplify the Constant Terms

Combine the constant terms:\[ 6 - 42 = -36 \].So, the expression simplifies to:\[ -36 - \sqrt{6} \].

Key Concepts

Binomial Multiplication

When dealing with polynomial expressions, binomial multiplication is fundamental. Imagine multiplying two binomials, such as \((a + b)(a + c)\). It helps to think of this as distributing over both terms in one binomial to each term in the other binomial. As often practiced in algebra, you will multiply:

• First terms
• Outer terms
• Inner terms
• Last terms

This technique is commonly referred to as the FOIL method (First, Outer, Inner, Last). By following this method, the expression \((\sqrt{6}+6)(\sqrt{6}-7)\) is expanded step-by-step to reach the simplest form.

Distributive Property

The distributive property is a crucial mathematical principle used in algebra to simplify expressions. It states that for any real numbers \(a\), \(b\), and \(c\): \[a(b + c) = ab + ac\]This rule indicates that you can distribute multiplication over addition or subtraction. For our example of \((\sqrt{6}+6)(\sqrt{6}-7)\), each term in the first binomial is multiplied by each term in the second binomial:

• \(\sqrt{6} \times \sqrt{6} = 6\)
• \(\sqrt{6} \times -7 = -7\sqrt{6}\)
• \(6 \times \sqrt{6} = 6\sqrt{6}\)
• \(6 \times -7 = -42\)

By applying the distributive property, binomial multiplication becomes straightforward, allowing you to simplify complex polynomial expressions.

Radical Expressions

A radical expression involves roots, such as square roots or cube roots. The term \(\sqrt{}\) represents a square root, which asks "what number squared equals this value?" In radical expressions like the one in our problem, we often deal with variables or numbers under a square root sign.
The simplification of radical expressions includes like radicals. When working with radicals, especially in binomial multiplications like our example, remember that only like radicals can be combined. This means that terms with the same radicand, such as \(-7\sqrt{6}\) and \(6\sqrt{6}\), can be combined to \(-\sqrt{6}\). Understanding radical expressions is key to mastering binomial multiplication and other complex algebraic operations.

Simplifying Expressions

Simplifying expressions is the process by which a mathematician rewrites an expression in its simplest form. Often, this involves combining like terms and reducing complex expressions.
In the exercise \((\sqrt{6}+6)(\sqrt{6}-7)\), after expanding using the distributive property, combining like terms is essential. Here, combining the radicals results in \(-\sqrt{6}\), and the constants \(6\) and \(-42\) combine to form \(-36\). Thus, the expression simplifies to \[-36 - \sqrt{6}\].
Through simplification, equations and expressions become more manageable, making it easier to handle subsequent mathematics problems.