Problem 89 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (3+\sqrt{7})(3-\sqrt{7}) $$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Identify the Expression Form

The given expression \((3 + \sqrt{7})(3 - \sqrt{7})\) is in the form of a difference of squares. The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

Using the formula from Step 1 where \(a = 3\) and \(b = \sqrt{7}\), substitute into the formula: \(a^2 - b^2 = (3)^2 - (\sqrt{7})^2\).

Step 3: Calculate Each Square

Calculate each term: \(3^2 = 9\) and \((\sqrt{7})^2 = 7\).

Step 4: Simplify the Expression

Subtract the two results from Step 3: \(9 - 7 = 2\). Thus, the simplified result of \((3+\sqrt{7})(3-\sqrt{7})\) is \(2\).

Key Concepts

Difference of Squares

The difference of squares is a special type of binomial that involves the subtraction of one square from another. This can be seen in the equation form

• \((a + b)(a - b) = a^2 - b^2\)

This form is quite common and is especially useful because it allows you to simplify expressions quickly. When you spot an expression like \((3+\sqrt{7})(3-\sqrt{7})\), your job becomes much easier since it fits the formula perfectly. Here, you identify that \(a = 3\) and \(b = \sqrt{7}\). By recognizing this form, calculations are reduced to squaring the two terms and then subtracting the results. This method swiftly cuts down on multiple steps, saving you time and effort.
It's important to remember that the difference of squares only works when you have exactly one positive and one negative term as shown. With practice, spotting these can become second nature.

Simplifying Expressions

Simplifying expressions is all about making a math problem neater or easier to solve. The goal is to express it in its simplest form without changing its value. To simplify an expression like \[(3 + \sqrt{7})(3 - \sqrt{7})\],you need to perform operations in a way that removes any complexities such as radicals or like terms.
Using the difference of squares in this context is incredibly handy because it takes the original expression involving a radical, such as \(\sqrt{7}\), and transforms it into a whole number, making it easier to work with. In the solution, once the formula is applied, you moved from a complex expression to a much simpler one: \(9 - 7\).
In essence, simplifying ensures that you're not dealing with anything more cumbersome than necessary, making it easier to grasp and solve. Training yourself to simplify expressions can help solve problems faster and with more clarity.

Multiplication of Binomials

When dealing with binomials, such as \((3 + \sqrt{7})(3 - \sqrt{7})\),it's essential to understand how their multiplication works. At first glance, this looks like a complicated task, but with the right techniques, it becomes straightforward.
For typical binomial multiplication, you might use the FOIL (First, Outer, Inner, Last) method. However, recognizing special forms, such as the difference of squares, allows you to bypass FOIL for a faster method.

• **First:** Recognize this as a difference of squares.
• **Substitute and simplify:** Directly apply the difference of squares formula, making your task much simpler.

This approach ensures that you efficiently and effectively calculate the product without unnecessary complexity.
Binomials often appear in algebra-related problems, and mastering their multiplication will boost your ability to tackle more advanced algebraic challenges with confidence.