Problem 37 - College Algebra Essentials

Question

$$\sqrt{8}+3 \sqrt{2}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{8}+3 \sqrt{2} \) is \(5 \sqrt{2}\).

Step 1: Simplify the square root

First, start by breaking down the \(\sqrt{8}\) into its simplest form that can be added up with \(3 \sqrt{2}\). The square root of 8 can be expressed as \(\sqrt{4*2}=2 \sqrt{2}\) because 4 is a perfect square.

Step 2: Combine like terms

Now, we can add \(2 \sqrt{2}\) with \(3 \sqrt{2}\) because they both have the same radical part (\(\sqrt{2}\)). The result is \(5 \sqrt{2}\).

Key Concepts

Radical Expressions

When we encounter expressions containing square roots, or any other roots, these are known as radical expressions. A radical expression includes both the radical symbol \( \sqrt{} \) and the quantity inside it, called the radicand. Simplifying these expressions involves finding an equivalent expression where the radicand has no perfect square factors. This process makes it easier to perform operations like addition or multiplication.

For example, simplifying \( \sqrt{8} \) involves identifying the perfect square factor of 8, which is 4, and rewriting it as \( \sqrt{4 \cdot 2} \) or \( 2\sqrt{2} \) because \( \sqrt{4} \) equals 2. The idea is to break down the original radical into simpler components that are more manageable to work with, whether for further simplification or for performing arithmetic operations with other radical terms.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. Numbers like 1, 4, 9, 16, and so on are perfect squares because they can be written as \(1^2\), \(2^2\), \(3^2\), \(4^2\), respectively. Recognizing perfect squares is vital for simplifying square roots, as a perfect square under a square root can be directly converted to its root.

For instance, the square root of 16 is easily simplified to 4 because 16 is a perfect square of 4. When simplifying radical expressions, the goal is to factor the radicand into a product that includes a perfect square, allowing us to remove the square root through simplification.

Combining Like Terms

The concept of combining like terms is used to simplify expressions by merging terms that have the same variable to the same power or the same radical part. In the case of radical expressions, like terms will have identical radicals.

For example, \(2 \sqrt{2}\) and \(3 \sqrt{2}\) are like terms because the radical part, \(\sqrt{2}\), is the same. These can be combined by simply adding or subtracting their coefficients: \(2 + 3\) times \(\sqrt{2}\) equals \(5 \sqrt{2}\). This combines both terms into a single, simplified expression. Understanding how to combine like terms is crucial in algebra to simplify expressions and solve equations effectively.