Problem 209 - Elementary Algebra

Question

In the following exercises, simplify. $$ 2 \sqrt{8}+6 \sqrt{8}-5 \sqrt{8} $$

Step-by-Step Solution

Answer

The simplified expression is \(3 \sqrt{8}\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. All terms in the expression involve the square root of 8. So we have: \(2 \sqrt{8}, 6 \sqrt{8}, -5 \sqrt{8}\)

Step 2: Combine Coefficients

Now, add or subtract the coefficients of the like terms. Given \(2 \sqrt{8}+6 \sqrt{8}-5 \sqrt{8}\), combine the coefficients as follows: \(2 + 6 - 5 = 3\).

Step 3: Write the Simplified Expression

Multiply the resulting coefficient by the common square root term: \(3 \sqrt{8}\). Therefore, the simplified form of the expression is \(3 \sqrt{8}\).

Key Concepts

Like Terms

In algebra, 'like terms' are terms that have the same variables and exponents. This makes them easy to combine through addition or subtraction. In the given exercise, every term involves \(\text{sqrt}(8)\). Because of this, they are called like terms.
To spot like terms:

• Look for identical variable parts.
• In this case, every term has the same square root, \(\sqrt{8}\).
• Identical variable parts mean the terms are 'like' and can be simplified together.

Identifying like terms is crucial because it simplifies complex expressions.

Coefficients

Coefficients are the numerical values attached to variables or radical expressions. They tell us how many times the term is multiplied. Let's take the terms \(2 \sqrt{8}\), \(6 \sqrt{8}\), and \(-5 \sqrt{8}\) from the exercise:

• \(2\) is the coefficient for \(2 \sqrt{8}\).
• \(6\) is the coefficient for \(6 \sqrt{8}\).
• \(-5\) is the coefficient for \(-5 \sqrt{8}\).

When combining like terms, you add or subtract these coefficients:

• Add \(2 + 6 - 5\).
• This results in \(3\), the combined coefficient.

Now, you end up with a simplified term: \(3 \sqrt{8}\).

Square Roots

Square roots are one of the basic operations in mathematics, often symbolized as \(\sqrt{}\). The square root of a number is a value that, when multiplied by itself, gives the original number.
Here is a quick breakdown:

• For example, \(\sqrt{8}\) is a number that, when squared, equals \(8\).
• Our terms all share this common square root (\(\sqrt{8}\)).
• This makes combination and simplification possible.

In the final simplified expression, \(3 \sqrt{8}\), the coefficient \(3\) tells us we have three sets of \(\sqrt{8}\). The concept of simplifying radical expressions revolves around combining like terms and coefficients in this way.