Problem 15 - Intermediate Algebra

Question

For Problems \(15-52\), find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. $$ \sqrt{2}(\sqrt{3}+\sqrt{5}) $$

Step-by-Step Solution

Answer

\( \sqrt{6} + \sqrt{10} \)

Step 1: Distribute the Radical

Apply the distributive property to the expression \( \sqrt{2}(\sqrt{3} + \sqrt{5}) \). This means you will multiply \( \sqrt{2} \) by each term inside the parentheses separately. Thus, the expression becomes \( \sqrt{2} \times \sqrt{3} + \sqrt{2} \times \sqrt{5} \).

Step 2: Calculate the Products

Simplify each term separately. The product of two square roots is another square root: \(\begin{align*}\sqrt{2} \times \sqrt{3} &= \sqrt{6}, \\sqrt{2} \times \sqrt{5} &= \sqrt{10}. \end{align*} \)

Step 3: Combine the Results

Combine the simplified terms from Step 2. The expression becomes \( \sqrt{6} + \sqrt{10} \).

Step 4: Simplest Radical Form

Check if the radical terms can be simplified further. Both \( \sqrt{6} \) and \( \sqrt{10} \) are already in their simplest forms based on their prime factors, so the final expression remains \( \sqrt{6} + \sqrt{10} \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It indicates that when you multiply a number by a sum, you can distribute the multiplication across each term individually. In our exercise, \( \sqrt{2} (\sqrt{3} + \sqrt{5}) \), we apply this principle by multiplying \( \sqrt{2} \) with each square root inside the parentheses, one at a time.

• First, multiply \( \sqrt{2} \times \sqrt{3} \).
• Then multiply \( \sqrt{2} \times \sqrt{5} \).

After these two separate multiplications, we get two products which we'll work with further.
Understanding this property is crucial because it helps in simplifying expressions and solving equations.
This technique isn't just limited to numbers; it's a versatile tool for any algebraic terms.

Square Roots

Square roots are numbers which, when multiplied by themselves, give the original number. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
In our exercise, we simplify expressions like \( \sqrt{2} \times \sqrt{3} \). Here we use the property of square roots:

• The product of square roots is equal to the square root of the product: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).

This is how we calculated \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \) in our solution.
Understanding how to manipulate square roots gives a better grip on simplifying expressions, making them easier to work with. Each square root in the final answer is checked to ensure it's in its simplest radical form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In algebra, these expressions can often include radicals, which are roots of numbers. For our problem, \( \sqrt{2} (\sqrt{3} + \sqrt{5}) \), involves both numbers and radicals creating a complex expression.

• Our goal is to simplify such expressions by performing operations like distribution and combining like terms.
• Here, we distributed \( \sqrt{2} \) across the terms in parentheses, then simplified the resulting products.

Such operations transform more complex expressions into manageable forms.
Algebra is about finding patterns and applying rules consistently. With radicals, the primary focus is to express each term in the simplest radical form to provide clarity and precision in solutions.