Problem 64 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 8 \sqrt{7}+2 \sqrt{7} $$

Step-by-Step Solution

Answer

The simplified expression is \( 10 \sqrt{7} \).

Step 1: Understand the Expression

The expression given is \( 8 \sqrt{7} + 2 \sqrt{7} \). Notice that both terms contain the same radical part, \( \sqrt{7} \). This implies that these terms are like terms.

Step 2: Factor Out the Common Term

Since \( 8 \sqrt{7} \) and \( 2 \sqrt{7} \) are like terms, you can factor out the common radical \( \sqrt{7} \) from each term.

Step 3: Perform Arithmetic on Coefficients

Combine the coefficients of the like terms: \( 8 + 2 = 10 \).

Step 4: Simplify the Expression

After factoring out and combining, the expression simplifies to \( 10 \sqrt{7} \). This is the simplest form.

Key Concepts

Factoring Out Common Terms

Factoring out common terms is a key skill in algebra to simplify expressions. When you have terms that share a common factor, such as a root, a variable, or a number, you can "factor out" this shared part. This means you will express the terms as a product of the common factor and something else.

For instance, in the expression \(8 \sqrt{7} + 2 \sqrt{7}\), both terms feature \(\sqrt{7}\). You treat \(\sqrt{7}\) as the common factor. You can write the expression as \((8 + 2) \sqrt{7}\).

This step might appear simple, but spotting the shared element is crucial. Once you pull out \(\sqrt{7}\), it makes simplifying much more manageable. It's like peeling off the outer wrapper to reveal what's the same underneath and easier to deal with.

Like Terms in Algebra

Understanding like terms is crucial when simplifying expressions in algebra. Like terms are terms in an expression that contain exactly the same variables raised to the same power. They might have different coefficients, but the variable part must match precisely.

In the expression \(8 \sqrt{7} + 2 \sqrt{7}\), both terms include the same root: \(\sqrt{7}\). This identical radical part makes them like terms. Like terms can be combined easily because they share this common element.

• The like terms here: \(8 \sqrt{7}\) and \(2 \sqrt{7}\)
• This similarity allows you to add or subtract their coefficients directly.

So whenever you encounter similar radical parts, check if they are like terms. If they are, they'll be your allies in the simplification process.

Arithmetic on Coefficients

Once you've identified and factored out the common term, the next step involves arithmetic on the coefficients. Coefficients are the numbers in front of the radicals or variables.

In the example \(8 \sqrt{7} + 2 \sqrt{7}\), after factoring out \(\sqrt{7}\), you focus on the coefficients \(8\) and \(2\). Treat these coefficients as regular numbers you can add, just like simple addition:

• Add them: \(8 + 2 = 10\)

The calculation here is straightforward but essential. This step ensures that you simplify the expression properly, resulting in \(10 \sqrt{7}\). By only adding the coefficients, you maintain the radical part unchanged, leading to an accurate simplification.