Problem 27 - 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. $$ \sqrt{8}(6+\sqrt{2}) $$

Step-by-Step Solution

Answer

The simplest form is \( 12\sqrt{2} + 4 \).

Step 1: Distribute the Expression

Distribute \( \sqrt{8} \) to each term within the parentheses. This is done by multiplying \( \sqrt{8} \) by each term: \[ \sqrt{8} \times 6 + \sqrt{8} \times \sqrt{2} \]

Step 2: Simplify Each Term

Simplify both terms. Firstly, consider the term \( \sqrt{8} \times 6 \). Since \( 6 \) is a constant, the expression becomes: \[ 6\sqrt{8} \] For the second term, \( \sqrt{8} \times \sqrt{2} \), use the property \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \) to simplify: \[ \sqrt{8 \times 2} = \sqrt{16} \] Since \( \sqrt{16} = 4 \), the second term simplifies to \( 4 \).

Step 3: Combine All Terms

Combine the simplified terms. The expression \( 6\sqrt{8} \) remains as is since there are no like terms to combine with it. Adding the second term, the expression becomes: \[ 6\sqrt{8} + 4 \]

Step 4: Simplify \( 6\sqrt{8} \) Further

Simplify the \( \sqrt{8} \) in the term \( 6\sqrt{8} \). Notice that \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2} \). Thus, \[ 6\sqrt{8} = 6 \times 2\sqrt{2} = 12\sqrt{2} \]

Step 5: Final Expression

Combine the simplified term \( 12\sqrt{2} \) with the constant term \( 4 \). Therefore, the simplest form is: \[ 12\sqrt{2} + 4 \]

Key Concepts

Radicals

A radical is a symbol that represents the root of a number. It's commonly denoted as the square root symbol \( \sqrt{} \). The number inside the symbol is called the radicand. For example, in \( \sqrt{8} \), 8 is the radicand. Radicals allow us to express roots of numbers, and simplifying them is key in algebra to ensure results are in their simplest form.

When simplifying radicals, we often look for perfect squares within the radicand, because they have whole number square roots. In our example, \( \sqrt{8} \) can be rewritten as \( \sqrt{4 \times 2} \), which simplifies further to \( 2\sqrt{2} \). Recognizing and simplifying radicals this way helps make calculations smoother and results simpler to interpret.

Distribution

Distribution is a method used to multiply a single term by each term in a separate group of terms, typically within parentheses. It involves "distributing" the multiplication across the terms. In our original exercise, this principle is applied in the expression \( \sqrt{8}(6 + \sqrt{2}) \).

To distribute \( \sqrt{8} \) through the terms in the parentheses, multiply it with each separate term:

• First multiply \( \sqrt{8} \times 6 \)
• Then multiply \( \sqrt{8} \times \sqrt{2} \)

This step is critical for breaking down expressions into simpler components before further simplification.

Multiplying Radicals

Multiplying radicals involves using an important property of roots: multiplying two square root expressions together allows you to multiply the radicands inside a single square root. Specifically, \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).

In our exercise, this property simplifies \( \sqrt{8} \times \sqrt{2} \) to \( \sqrt{16} \), which is easy to evaluate since \( 16 \) is a perfect square, giving a result of \( 4 \).

• Recognize perfect squares: They simplify beautifully.
• Use multiplication of radicands to keep expressions tidy.

These steps ensure radical expressions are managed and simplified effectively through multiplication.

Properties of Radicals

Radicals follow specific mathematical properties that allow for different operations, such as simplification and multiplication. These properties are fundamental to making radicals manageable within various expressions:

• **Product Property:** \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \)
• **Simplification Property:** Take out perfect squares from the radicand, \( \sqrt{4 \times 2} \rightarrow 2\sqrt{2} \)

In our final expression, we used these properties to achieve the simplest form: \( 12\sqrt{2} + 4 \). Understanding and applying these properties efficiently provides consistent and simplified results in algebra.