Problem 287 - Elementary Algebra

Question

In the following exercises, simplify. $$ (6+5 \sqrt{n})(11+3 \sqrt{n}) $$

Step-by-Step Solution

Answer

\( 66 + 73 \sqrt{n} + 15n \)

Step 1 - Apply the distributive property

Use the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand the expression \( (6+5 \sqrt{n})(11+3 \sqrt{n}) \).

Step 2 - Multiply the first terms

Multiply the first terms: \( 6 \times 11 = 66 \)

Step 3 - Multiply the outer terms

Multiply the outer terms: \( 6 \times 3 \sqrt{n} = 18 \sqrt{n} \)

Step 4 - Multiply the inner terms

Multiply the inner terms: \( 5 \sqrt{n} \times 11 = 55 \sqrt{n} \)

Step 5 - Multiply the last terms

Multiply the last terms: \( 5 \sqrt{n} \times 3 \sqrt{n} = 15n \). Recall that \( \sqrt{n} \times \sqrt{n} = n \)

Step 6 - Combine like terms

Add all the products together: \( 66 + 18 \sqrt{n} + 55 \sqrt{n} + 15n \). Combine the \( \sqrt{n} \) terms: \( 18 \sqrt{n} + 55 \sqrt{n} = 73 \sqrt{n} \)

Step 7 - Write the final expression

Combine everything into one expression: \( 66 + 73 \sqrt{n} + 15n \)

Key Concepts

distributive property

The distributive property in algebra is a very useful tool. It allows us to simplify expressions and solve equations that involve multiple terms. In this particular problem, we use the distributive property to expand the expression ewline ewline (6+5 \/sqrt{n})(11+3 \/sqrt{n}). Essentially, it means multiplying each term inside the first parenthesis by every term in the second parenthesis. You may also hear this concept referred to as the FOIL method, which stands for First, Outer, Inner, Last. Here’s a quick rundown: ewline

• First: Multiply the first terms in each parenthesis. For our problem, that is \(6 \times 11 = 66\).
• Outer: Multiply the outer terms in the expression. In this case, \(6 \times 3 \/sqrt{n} = 18 \/sqrt{n}\).
• Inner: Multiply the inner terms. Here it is \(5 \/sqrt{n} \times 11 = 55 \/sqrt{n}\).
• Last: Finally, multiply the last terms in each binomial: \(5 \/sqrt{n} \times 3 \/sqrt{n} = 15n\). Remember that \( \/sqrt{n} \times \/sqrt{n} = n\), which simplifies our expression nicely.

Taking the distributive property step-by-step ensures that we do not overlook any terms. Once every term has been distributed, we can move on to combining like terms.

multiplying radicals

Multiplying radicals can seem intimidating at first, but it’s quite straightforward. When we multiply numbers under the radical sign, we multiply them just as if they were regular numbers. However, it's essential to remember a key rule: the square root of a product is the same as the product of the square roots, which we write as:ewlineewline \[\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\] ewlineewline When multiplying radicals that are the same, like \(5 \/sqrt{n} \times 3 \/sqrt{n}\), we also follow another important principle: \( \/sqrt{n} \times \/sqrt{n} = n\). Here’s how it works in our problem:ewline

• First, when we multiplied the first and outer terms in our example, we had simple multiplications involving one radical term, so we got \(18 \/sqrt{n}\) and \(55 \/sqrt{n}\).
• Next, we multiplied the inner terms \(5 \/sqrt{n} \) and the \(11\), which resulted in another simple calculation, giving us \(55 \/sqrt{n}\).
• Lastly, we dealt with multiplying \(5 \/sqrt{n} \times 3 \/sqrt{n}\), and using \( \/sqrt{n} \times \/sqrt{n}=n\), we found our answer to be \(15n\).

Understanding these steps is essential when handling more complex algebraic expressions. It ensures that you accurately simplify each part of the equation before combining like terms.

combining like terms

Combining like terms is the process of simplifying expressions by summing terms that have identical variables raised to the same power. This concept is crucial to reducing expressions to their simplest form and making them easier to solve or evaluate. In our example, after we've distributed and multiplied all terms, we end up with several terms that need combining. Here’s how we proceed step-by-step:

• We start with the expanded form: \(66 + 18 \/sqrt{n} + 55 \/sqrt{n} + 15n\).
• First, identify the like terms among them. Like terms are terms that contain the same variable raised to the same power.
• In this problem, \(18 \/sqrt{n} \) and \(55 \/sqrt{n}\) are like terms as both contain \( \/sqrt{n}\). We add these together: \( 18 \/sqrt{n} + 55 \/sqrt{n}= 73 \/sqrt{n} \).
• Now, include this result back into the equation and combine all simplified terms to get the final expression: \(66 + 73 \/sqrt{n} + 15n\).

In combining like terms, we make sure our final answer is concise and easy to understand. This process is vital, especially when dealing with more complicated algebraic expressions.