Problem 13 - Algebra 2

Question

Multiply. $$ (3+\sqrt{5})(1+\sqrt{5}) $$

Step-by-Step Solution

Answer

The solution is \(8 + 4\sqrt{5}\).

Step 1: Multiplication of the First Terms

Multiply the first terms in each binomial: \(3 * 1 = 3\).

Step 2: Multiplication of the Outer Terms

Next, multiply the outer terms: \(3 * \sqrt{5}\) giving \(3\sqrt{5}\).

Step 3: Multiplication of the Inner Terms

Then multiply the inner terms: \(\sqrt{5} * 1\) giving \(\sqrt{5}\).

Step 4: Multiplication of the Last Terms

Lastly, multiply the last terms: \(\sqrt{5} * \sqrt{5}\) gives \(5\).

Step 5: Summation of the Results

Now add all these results together: \(3 + 3\sqrt{5} + \sqrt{5} + 5\). Combining like terms gives \(3\sqrt{5} + \sqrt{5} = 4\sqrt{5}\) and \(3 + 5 = 8\). So the final answer is \(8 + 4\sqrt{5}\).

Key Concepts

Binomial Multiplication

Binomial multiplication is a critical algebraic concept often necessary when simplifying expressions or solving equations. A binomial is an expression with two terms, such as

• (3 + \( \sqrt{5} \))
• (1 + \( \sqrt{5} \))

Multiplying these requires a method often remembered by the acronym FOIL, which stands for:

• First
• Outer
• Inner
• Last

Each represents a specific pair of terms from each binomial that you'll multiply together. Here's how it works in the given exercise:- **First Terms:** Multiply the first terms of each binomial (3 and 1), resulting in 3.- **Outer Terms:** Multiply the terms on the outer ends (3 and \( \sqrt{5} \)), resulting in \( 3\sqrt{5} \).- **Inner Terms:** Multiply the inside terms (\( \sqrt{5} \) and 1), giving \( \sqrt{5} \).- **Last Terms:** Multiply the last terms of each binomial (\( \sqrt{5} \) and \( \sqrt{5} \)), resulting in 5.After you've multiplied these terms, you add them together to form a new expression. Understanding binomial multiplication allows you to simplify and solve more complex algebraic expressions effectively.

Radical Expressions

Radical expressions involve roots, most commonly square roots, represented by the radical sign \( \sqrt{} \). In the given exercise, radical expressions are prominent in both binomials. Understanding how to manipulate these expressions is crucial for algebra.A key aspect of handling radical expressions is property utilization, which helps simplify problems:

• **Product of Radicals:** \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \)
• **Rationalization:** Working with radicals often requires rationalizing, or removing, the radical from a denominator, which isn't fully covered here but is good to know.

In multiplying radical expressions, notice that:- When \( \sqrt{5} \) is multiplied by another radical term like \( \sqrt{5} \), the result is no longer a radical as the expression \( \sqrt{25} \) simplifies to 5.Being comfortable with radicals integrates nicely with other algebraic operations, helping you combine terms and eventually simplify expressions.

Combining Like Terms

After multiplying through binomial expressions and dealing with any radicals, you'll often need to combine like terms. This step is vital for simplifying the expression to its cleanest form. Like terms are terms that have the same variables raised to the same powers. It's crucial to recognize these terms to perform the operation correctly.Here's how combining like terms works in the provided solution:- Look for terms with the same radical. For example, \( 3\sqrt{5} \) and \( \sqrt{5} \) can be added together because they both contain the radical \( \sqrt{5} \). This gives \( 4\sqrt{5} \).- Similarly, constant terms - numbers without radicals or variables - like 3 and 5 can be combined, resulting in 8.Combining like terms simplifies the expression and makes it easier to interpret and use in further calculations. Recognizing patterns and applying consistent rules will help you to manage more complex expressions effectively.