Problem 25 - 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. $$ \sqrt{2}(2+\sqrt{2}) $$

Step-by-Step Solution

Answer

The product in simplest form is \( 2\sqrt{2} + 2 \).

Step 1: Distribute the Square Root

Apply the distributive property to the expression \( \sqrt{2}(2+\sqrt{2}) \). This means you multiply \( \sqrt{2} \) by each term inside the parentheses separately. The expression becomes:\[ \sqrt{2} \times 2 + \sqrt{2} \times \sqrt{2} \]

Step 2: Simplify the Multiplications

Now, calculate each term:- \( \sqrt{2} \times 2 = 2\sqrt{2} \)- \( \sqrt{2} \times \sqrt{2} = \sqrt{2 \times 2} = \sqrt{4} = 2 \)So the expression becomes:\[ 2\sqrt{2} + 2 \]

Step 3: Combine the Terms

The expression \( 2\sqrt{2} + 2 \) consists of a term with a radical and a constant term. Since there are no like terms to combine further, this is already the simplest form.Thus, the simplest form of the product is \( 2\sqrt{2} + 2 \).

Key Concepts

Understanding the Distributive Property with Radicals

When working with expressions involving radicals, the distributive property is a powerful tool. Imagine you have an expression like \( \sqrt{2}(2+\sqrt{2}) \). The distributive property allows us to break down this expression by multiplying each term inside the parentheses separately by \( \sqrt{2} \). This step is crucial because it ensures that each component is considered and allows for further simplification.

Here's how it works:

• You multiply \( \sqrt{2} \) by 2, which yields \( \sqrt{2} \times 2 \).
• Then, multiply \( \sqrt{2} \) by \( \sqrt{2} \), which gives \( \sqrt{2} \times \sqrt{2} \).

By applying the distributive property in this manner, we transform our initial expression into terms that are simplifiable. This approach sets the stage for the next steps in the simplification process, ensuring a clear path to the simplest form.

Multiplying Radicals

After distributing, the next step is to handle the multiplication of radicals. Radicals can be multiplied much like integers, but with the added step of recognizing the products inside the radical. First, when you have \( \sqrt{2} \times 2 \), we simply place the constant outside, giving us \( 2\sqrt{2} \). The radical \( \sqrt{2} \) remains attached to the number 2.

Now, multiplying \( \sqrt{2} \times \sqrt{2} \) results in \( \sqrt{4} \). This happens because under the radical, numbers multiply just like whole numbers do: \( 2 \times 2 = 4 \). Recognizing that \( \sqrt{4} = 2 \) allows this term to simplify immediately.

• It's vital to remember that \( \sqrt{a} \times \sqrt{a} = a \) is a guiding formula.
• This simplification reduces clutter in our expression and makes the final combination of terms much clearer.

Mastering the operation of radicals is essential for tackling more complex expressions.

Simplifying Expressions Involving Radicals

The final task in working with our expression is to bring it all together. After distributing and multiplying, our expression is already broken down into \( 2\sqrt{2} + 2 \).In this form, no further simplification is possible because the terms are unlike; one term includes a radical while the other is a constant.

• When combining terms, check if like terms exist. Like terms would mean both have radicals involving the same base or neither having radicals.
• In this context, \( 2\sqrt{2} \) and 2 cannot be combined because one is a radical term and the other is a constant.

Simplifying expressions means rewriting them in a form that is easier to understand or evaluate, ensuring clarity when dealing with operations later. This approach ultimately leads to the simplest form of the expression, concluding the problem-solving process.