Problem 31 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (9+\sqrt{2 b})(1+\sqrt{2 b}) $$

Step-by-Step Solution

Answer

\(2b + 9 + 10\sqrt{2b}\)

Step 1: Expand the Product

To simplify the expression \((9+\sqrt{2b})(1+\sqrt{2b})\), we need to use the distributive property to expand the product. This involves multiplying each term in the first binomial by each term in the second binomial:\((9+\sqrt{2b})(1+\sqrt{2b}) = 9 \cdot 1 + 9 \cdot \sqrt{2b} + \sqrt{2b} \cdot 1 + \sqrt{2b} \cdot \sqrt{2b}.\)

Step 2: Calculate the Products

Now, calculate each of the products obtained from the expansion:- \(9 \cdot 1 = 9\)- \(9 \cdot \sqrt{2b} = 9\sqrt{2b}\)- \(\sqrt{2b} \cdot 1 = \sqrt{2b}\)- \(\sqrt{2b} \cdot \sqrt{2b} = 2b\)

Step 3: Combine Like Terms

Combine all the calculated terms from Step 2:\[ 9 + 9\sqrt{2b} + \sqrt{2b} + 2b. \] Combine the like terms \(9\sqrt{2b}\) and \(\sqrt{2b}\):\[ 9 + 10\sqrt{2b} + 2b. \]

Step 4: Write the Final Expression

The simplest form of the product \((9+\sqrt{2b})(1+\sqrt{2b})\) is:\[ 2b + 9 + 10\sqrt{2b}. \]

Key Concepts

Binomial Expansion

Binomial expansion is an essential concept that helps us simplify expressions involving products of binomials. In the mathematical world, a binomial is simply an expression that consists of two terms connected by a plus or minus sign. For instance, in our exercise, we have two binomials:

• First binomial: (9 + \( \sqrt{2b} \))
• Second binomial: (1 + \( \sqrt{2b} \)).

When we multiply two binomials together, we use a series of multiplications to expand the expression. This is known as binomial expansion. The process involves multiplying each term of the first binomial with each term of the second. This method might seem complex, but it allows us to break down expressions into simpler terms. Thus, by using the structure of multiplication, we simplify the expression:(9 + \( \sqrt{2b} \)) \( \cdot \) (1 + \( \sqrt{2b} \)) to obtain an expanded form:9 \( \cdot \) 1 + 9 \( \cdot \sqrt{2b} \) + \( \sqrt{2b} \) \( \cdot \) 1 + \( \sqrt{2b} \) \( \cdot \sqrt{2b} \). This highlights the importance of understanding binomial expansion, as it is a cornerstone of simplifying more complex expressions.

Distributive Property

The distributive property is a fundamental algebraic principle that plays a crucial role in simplifying expressions. The property essentially states that when we distribute a term over terms within a parenthesis, we need to multiply that term by each term inside the parenthesis separately. In our exercise, the distributive property is used to expand the binomials:

• 9 \( \cdot \) 1 + 9 \( \cdot \sqrt{2b} \)
• \( \sqrt{2b} \) \( \cdot \) 1 + \( \sqrt{2b} \) \( \cdot \sqrt{2b} \)

Through these operations, each term from the first binomial is distributed and multiplied by every term in the second binomial. This is an effective strategy to break down and solve complex mathematical problems. By understanding and applying this property, students can navigate through the most intricate algebraic expressions with confidence. Ultimately, these calculations allow us to simplify the expression from its expanded form to simplify our final expressions effectively.

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and beyond. In the context of our exercise, the term \( \sqrt{2b} \) is a radical, specifically a square root. Understanding radicals and how to manipulate them is essential for simplification. When dealing with radicals, it's important to remember a few basic principles:

• If you multiply a radical by itself, it "undoes" the radical, yielding the radicand (the expression inside the radical sign). For example, \( \sqrt{2b} \cdot \sqrt{2b} = 2b \).

In Step 2 of our solution, when we calculated \( \sqrt{2b} \cdot \sqrt{2b} \), we simplified it to 2b. Combining the like radical terms also requires attention. In this scenario, the like terms are 9\( \sqrt{2b} \) and \( \sqrt{2b} \) after the expansion. When combined, they give us 10\( \sqrt{2b} \). Having a firm grasp of how to manage radicals enables students to simplify and manage expressions accurately, a crucial skill in further mathematical endeavors.