Problem 44 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is 3.

Step 1: Recognize the Expression Type

The expression \((\sqrt{7}-2)(\sqrt{7}+2)\) is in the form of a difference of squares. A difference of squares formula is given by \((a-b)(a+b) = a^2 - b^2\). In this case, \(a = \sqrt{7}\) and \(b = 2\).

Step 2: Apply the Difference of Squares Formula

Using the formula for the difference of squares, substitute \(a\) and \(b\) into \(a^2 - b^2\). Here, you will calculate \((\sqrt{7})^2 - 2^2\).

Step 3: Calculate the Squares

First, calculate \((\sqrt{7})^2\), which is 7, because the square of a square root results in the number under the square root. Then, calculate \(2^2\), which is 4.

Step 4: Subtract the Results

Now subtract the result of \(2^2\), which is 4, from the result of \((\sqrt{7})^2\), which is 7, to find the simplified form of the expression: \(7 - 4 = 3\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a very useful pattern that simplifies the multiplication of certain binomials. When faced with terms like \[(a-b)(a+b)\]this is known as the difference of squares pattern. It tells us that the product of these two expressions results in:\[a^2 - b^2\]This is because the middle terms resulting from such multiplication cancel each other. In our original exercise, the terms \[( \sqrt{7} - 2 )(\sqrt{7} + 2)\] fit this pattern perfectly. Recognizing this pattern allows you to bypass the distribution process and jump directly to the solution by identifying your values for \(a\) and \(b\). In this case, \(a = \sqrt{7}\) and \(b = 2\), so our difference of squares formula simplifies the expression to \[ \sqrt{7}^2 - 2^2 \],leading you directly to the result.

Simplest Form

The simplest form of an expression is the version of it that is most concise and straightforward, without losing any information. When dealing with numbers and variables, the simplest radical form refers to a radical expression that has been reduced as much as possible.Here's a step-by-step breakdown of simplifying with radicals:

• First, evaluate expressions inside radicals, such as \((\sqrt{7})^2 = 7\).
• Next, perform any other operations necessary, like multiplication or subtraction, outside of the radical.
• Finally, combine and simplify whenever possible, ensuring there are no square numbers left inside the radical.

In our example, after applying the difference of squares formula, we calculated \[7 - 4,\] which equals \(3\). This answer is already in its simplest form.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Simplifying radical expressions requires understanding how to handle these roots properly. The main goals are to:

• Reduce any square roots by removing perfect squares.
• Ensure no fractions remain under the radical sign—only integers should be present.
• Present the results clearly and efficiently for easy interpretation.

In the specific exercise involving \[( \sqrt{7} - 2 )(\sqrt{7} + 2),\]although each square root expression seems complex at first, once you apply the difference of squares formula, the radicals simplify into regular integer operations. Radicals like \(\sqrt{7}\) don't simplify into further smaller numbers inside the expression themselves. Through subtraction and calculating squares, the solution process yields simple whole numbers, thereby eliminating the need to deal with roots in the final answer.