Problem 316 - Elementary Algebra

Question

Use the binomial square pattern to simplify \((3+\sqrt{2})^{2} .\) Explain all your steps.

Step-by-Step Solution

Answer

The simplified form of \( (3 + \sqrt{2})^{2} \) is \ 11 + 6\sqrt{2} \.

Step 1: Recall the Binomial Square Pattern

The binomial square pattern is given by \[\begin{equation}(a + b)^{2} = a^{2} + 2ab + b^{2}.\end{equation}\]In this problem, we are given \[\begin{equation}(3 + \sqrt{2})^{2}.\end{equation}\]Here, \a = 3 and \b = \sqrt{2}.

Step 2: Square the First Term

Calculate \( a^{2} \), where \ a = 3. \[\begin{equation} a^{2} = 3^{2} = 9. \end{equation}\]

Step 3: Calculate Twice the Product of the Two Terms

Calculate \( 2ab \), where \ a = 3 \ and \ b = \sqrt{2}. \[\begin{equation}2ab = 2(3)(\sqrt{2}) = 6\sqrt{2}.\end{equation}\]

Step 4: Square the Second Term

Calculate \( b^{2} \), where \ b = \sqrt{2}. \[\begin{equation} b^{2} = (\sqrt{2})^{2} = 2.\end{equation}\]

Step 5: Combine All the Terms

Add all the calculated terms together: \[\begin{equation}(3 + \sqrt{2})^{2} = 9 + 6\sqrt{2} + 2. \end{equation}\] Combine like terms: \[\begin{equation}= 11 + 6\sqrt{2}.\end{equation}\]

Key Concepts

Simplifying Binomials

Simplifying binomials is a crucial concept in algebra. A binomial is an expression with two terms, usually separated by a plus (+) or minus (−) sign. In our case, the binomial expression is \(3 + \sqrt{2}\). When asked to simplify the square of a binomial, we use the binomial square pattern. The pattern is given by: \((a + b)^2 = a^2 + 2ab + b^2\). This helps us expand and simplify binomials effectively by following some known algebraic steps.

Exponents

Exponents are a fundamental part of algebra and are used to indicate that a number (the base) is multiplied by itself a certain number of times. In the problem, we are squaring the binomial \(3 + \sqrt{2}\). Squaring means raising the binomial to the power of 2. When simplifying, remember the basic exponent rule: when a number is raised to the power of 2, like \((3^{2})\), it means \(3 \times 3 = 9\). The same applies when squaring \(\root{2}\) to get 2.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operators (like +, -, ×, ÷). Our given expression, \(3 + \sqrt{2}\), is an example of a binomial, which is a simple algebraic expression. Through this exercise, you're learning to manipulate and simplify more complex algebraic expressions by expanding and combining like terms using known algebraic formulas and patterns such as the binomial square pattern. This understanding is vital for solving more intricate problems in algebra.

Basic Algebra

Basic algebra involves understanding and applying principles to manipulate algebraic expressions just like the binomial square pattern in our exercise. This foundational knowledge helps recognize patterns, follow rules for arithmetic operations, and combine like terms. For instance, in this exercise, you see how each term from the expanded binomial square pattern corresponds to a simple arithmetic operation: squaring each term separately, multiplying them, and then adding the results. These steps highlight the systemic process of algebra evaluation.