Problem 71 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

-7

Step 1: Identify the Expression Format

The given expression \[(\sqrt{2}+3)(\sqrt{2}-3)\] follows the format of \[(a + b)(a - b)\] where \(a = \sqrt{2}\) and \(b = 3\). This format is known as the difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \[(a+b)(a-b) = a^2 - b^2\]. Substituting \(a = \sqrt{2}\) and \(b = 3\) into the formula, we get: \[ (\sqrt{2})^2 - 3^2. \]

Step 3: Calculate Each Square

Calculate the square of each term:- \((\sqrt{2})^2 = 2\), because squaring a square root eliminates the square root.- \(3^2 = 9\).

Step 4: Simplify the Expression

Substitute the calculated values back into the simplified expression:\[ 2 - 9. \]Subtract to get the final simplified result.

Step 5: Calculate the Final Result

Perform the subtraction:\[ 2 - 9 = -7. \]Thus, the expression simplifies to \(-7\).

Key Concepts

Simplification of Expressions

Simplifying an expression involves reducing it to its simplest form without changing its value. This simplifies computations and makes it easier to understand the expression's behavior. In our original exercise, the expression \[(\sqrt{2}+3)(\sqrt{2}-3)\]is simplified using the difference of squares formula. Here are some steps you often follow when simplifying expressions:

• Identify common patterns (like difference of squares, perfect square trinomials, etc.).
• Apply the appropriate algebraic identities.
• Perform arithmetic operations, such as adding or subtracting like terms.

It's essential to practice these techniques so you can quickly identify and simplify expressions in various forms.
Recognizing patterns like \( (a+b)(a-b) = a^2 - b^2 \) is a key skill, as it allows us to simplify without extensive multiplication. In this case, it helped reduce the expression to a straightforward subtraction: \(2 - 9 = -7\).

Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes both positive and negative numbers, zero, whole numbers, fractions, and irrational numbers. Variables in math exercises, like the one we're working on, often represent real numbers because they are tangible and relatable.
Understanding real numbers is fundamental as:

• They allow for precise and varied mathematical operations, including addition, subtraction, multiplication, and division.
• They include integers, rational numbers (like fractions), and irrational numbers (such as \(\sqrt{2}\)).

In our exercise, both \(\sqrt{2}\) and \(3\) from our original expression \[(\sqrt{2}+3)(\sqrt{2}-3)\] are real numbers. The result of our simplification process \(-7\) is also a real number. Even negative results like \(-7\) fall within the real numbers skope, emphasizing the all-encompassing nature of this set.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. It's the reverse operation of squaring a number. For instance, the square root of a number \(n\) is \( \sqrt{n} \).
Square roots play a significant role in simplifying expressions, as seen in our example. Here are a few key points about square roots:

• The square root of a perfect square is always an integer (e.g., \(\sqrt{9} = 3\)).
• Square roots of non-perfect squares, like \(\sqrt{2}\), are irrational numbers.
• When you square a square root, they cancel out (e.g., \((\sqrt{2})^2 = 2\)).

In our exercise, simplifying \((\sqrt{2}+3)(\sqrt{2}-3)\) involves calculating the square \((\sqrt{2})^2\), which returns the base number \(2\). This step is essential because it allows us to utilize the difference of squares identity correctly, leading to the simplification final result.