Problem 34 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{7}-2)(\sqrt{7}-8)\)

Step-by-Step Solution

Answer

The product is \(23 - 10\sqrt{7}\).

Step 1: Identify the Expression

The expression given is \((\sqrt{7} - 2)(\sqrt{7} - 8)\). We are multiplying two binomials.

Step 2: Apply the Distributive Property

Use the FOIL method (First, Outer, Inner, Last) to expand the binomials: - First: \(\sqrt{7} \times \sqrt{7} = 7\)- Outer: \(\sqrt{7} \times (-8) = -8\sqrt{7}\)- Inner: \((-2) \times \sqrt{7} = -2\sqrt{7}\)- Last: \((-2) \times (-8) = 16\)

Step 3: Combine Like Terms

Combine the results from the FOIL method:\(7 - 8\sqrt{7} - 2\sqrt{7} + 16\). Combine the constant terms and then the radical terms:\(7 + 16 = 23\)\(-8\sqrt{7} - 2\sqrt{7} = -10\sqrt{7}\)

Step 4: Simplify the Expression

The expression simplifies to:\(23 - 10\sqrt{7}\)

Key Concepts

Distributive Property in Algebra

The distributive property in algebra is a fundamental principle that comes in handy, especially when you're dealing with expressions like \((\sqrt{7} - 2)(\sqrt{7} - 8)\). This property allows us to multiply a single term by two or more terms inside a parenthesis. The distributive property states that \(a(b + c) = ab + ac\).
To tackle this operation, we start by understanding that we're essentially distributing each term in one binomial to every term in the other binomial. This approach simplifies our computation, making it a preferred method for expanding expressions. For instance, in the original problem, we express each multiplication separately:

• The term \(\sqrt{7}\) is multiplied by both \(\sqrt{7}\) and \(-8\), and
• The term \(-2\) is multiplied by \(\sqrt{7}\) and \(-8\).

Using this process ensures that all parts of the expression are properly accounted for, leading to a correct expanded form.

FOIL Method

The FOIL method is a specific application of the distributive property for multiplying two binomials. The term "FOIL" is an acronym that helps us remember the order of multiplying the terms: First, Outer, Inner, and Last.
Here's how it works in the context of our expression \((\sqrt{7} - 2)(\sqrt{7} - 8)\):

• First: Multiply the first terms in each binomial: \(\sqrt{7} \times \sqrt{7} = 7\).
• Outer: Multiply the outer terms: \(\sqrt{7} \times -8 = -8\sqrt{7}\).
• Inner: Multiply the inner terms: \(-2 \times \sqrt{7} = -2\sqrt{7}\).
• Last: Multiply the last terms in each binomial: \(-2 \times -8 = 16\).

The result of using FOIL is a four-term expression that combines the results of these multiplications. The great thing about the FOIL method is that it is straightforward and systematic, making it an excellent tool for expanding binomials effectively.

Combining Like Terms

Once we've expanded our expression using the FOIL method, the next critical step is simplifying it by combining like terms. Like terms refer to the terms in an expression that have the same variable raised to the same power, or in this case, the same radical.
In our example, after applying FOIL to \((\sqrt{7} - 2)(\sqrt{7} - 8)\), we obtain \(7 - 8\sqrt{7} - 2\sqrt{7} + 16\).
Here's how we combine like terms:

• Identify the constant terms: These are \(7\) and \(16\). When combined, they yield \(23\).
• Identify the radical terms: These are \(-8\sqrt{7}\) and \(-2\sqrt{7}\). They share the same radicand, \(\sqrt{7}\), which allows us to combine them into \(-10\sqrt{7}\).

Combining like terms is a crucial step that helps to condense an expression into its simplest form, making it more manageable and easier to understand. It brings the final expression to be \(23 - 10\sqrt{7}\), providing a clean and complete solution.