Problem 14 - Algebra 2

Question

Multiply. $$ (2+\sqrt{7})(1+3 \sqrt{7}) $$

Step-by-Step Solution

Answer

The multiplication of the given binomials results in \(23 + 7\sqrt{7}\).

Step 1: Apply the distributive property

Apply the distributive property (FOIL), which stands for: First terms, Outer terms, Inner terms, Last terms. This will give us: \(2*1 + 2*3\sqrt{7} + \sqrt{7}*1 + \sqrt{7}*3\sqrt{7}\).

Step 2: Simplify the obtained terms

Now, simplify all the terms that we obtained in step 1. This gives us: \(2 + 6\sqrt{7} + \sqrt{7} + 21\).

Step 3: Combine like terms

Next, combine the like terms together. This leaves us with \(23 + 7\sqrt{7}\).

Key Concepts

Distributive Property in Algebra

The distributive property is a crucial algebraic concept used to simplify expressions, particularly when multiplying two binomials. It helps break down a more complex expression into manageable parts. In the original exercise, the expression

• (2 + \(\sqrt{7}\))(1 + 3\(\sqrt{7}\))

applies this property effectively. Using the distributive property here is achieved by multiplying each term in the first binomial by each term in the second binomial, often remembered by the acronym FOIL:

• First: Multiply the first terms: \(2 \times 1\)
• Outer: Multiply the outer terms: \(2 \times 3\sqrt{7}\)
• Inner: Multiply the inner terms: \(\sqrt{7} \times 1\)
• Last: Multiply the last terms: \(\sqrt{7} \times 3\sqrt{7}\)

This process results in four separate products that will be summed together. Each step gives us new terms to work with in the expression, which lays the groundwork for the next stage: simplification.

Simplifying Expressions

Simplifying expressions involves reducing expressions into their simplest form. After using the distributive property, we get an expression consisting of several terms:

• \(2 + 6\sqrt{7} + \sqrt{7} + 21\)

Our goal is to condense and streamline these terms without changing the expression's value.
First, perform any immediate calculations, such as multiplying constants and combining radicals. Here, when we compute \(\sqrt{7} \times 3\sqrt{7}\), we've already incorporated some of this idea. The calculation results in multiplying under the radical, \(\sqrt{21}\), leading to 21, since \(\sqrt{7} \times \sqrt{7} = 7\).
Check every term for possible simplification opportunities, ensuring all parts of the expression are as concise as possible. Successfully simplifying them makes it easier to solve or further manipulate the expression in later calculations.

Combining Like Terms

Combining like terms is essential for simplifying expressions in algebra. It involves merging terms that have the same variable parts. In our present example

• \(2 + 6\sqrt{7} + \sqrt{7} + 21\)

we look for terms with identical components to combine them into a single term.
Here, \(2\) and \(21\) are constants, and by combining them, we get \(23\). Similarly, \(6\sqrt{7}\) and \(\sqrt{7}\) both involve \(\sqrt{7}\). Adding these coefficients, we obtain \(7\sqrt{7}\).
The result of combining like terms in our expression leads us to a more simplified form:

• \(23 + 7\sqrt{7}\)

This step is crucial since it helps in reducing the expression to a form where no further simplification of similar terms is possible, making evaluation or additional operations much more manageable.