Problem 58 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{18}\)

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{18}\) is \(3\sqrt{2}\).

Step 1: Identify the Factors

First, identify the factors of the number under the square root. For \(18\), these can be expressed as a product of two integers, such as \(18 = 9 \times 2\).

Step 2: Break Down the Radicals

Utilize the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Applying this to our expression, we have \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2}\).

Step 3: Simplify the Square Roots

Simplify any perfect squares. Since \(\sqrt{9} = 3\), substitute it back, giving \(3 \times \sqrt{2}\).

Step 4: Final Result

Thus, the expression \(\sqrt{18}\) simplifies to \(3\sqrt{2}\).

Key Concepts

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because when 3 is multiplied by itself (3 x 3), it equals 9. In mathematical terms, if you have \( = x^2\), then the square root of \(\) is \(\).When simplifying square roots, you want to express it in its simplest form. This involves breaking down the number into its prime factors and identifying any perfect squares. Once identified, you use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This property is super helpful as it allows you to simplify complex square roots by dealing with simpler parts.When you look at square roots in radical expressions like \(\sqrt{18}\), your goal is to simplify it as much as possible by identifying perfect squares within it. We'll explore that in the next sections.

Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. Common examples are 1, 4, 9, 16, 25, etc. These numbers are significant when simplifying radicals because they can be easily extracted from the square root, making the expression simpler.For instance, in the expression \(\sqrt{18}\):

• First, find the factors of 18, which are 9 and 2.
• Recognize that 9 is a perfect square because \(3 \times 3 = 9\).
• Apply this knowledge to simplify \(\sqrt{18}\) to \(3\sqrt{2}\).

By identifying and extracting the square root of perfect squares, you change the expression to a simpler form. This makes it easier to work with in further calculations or algebraic expressions.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations like addition, subtraction, multiplication, and division. Square roots often appear in these expressions and simplifying radicals plays a key role in making them easier to manage.Consider the simplified form of \(\sqrt{18}\) which is \(3\sqrt{2}\). Though it looks different, it essentially represents the same value, just in a simpler and more workable format. When you insert this into algebraic expressions, having it simplified as \(3\sqrt{2}\) often allows you to integrate it into larger expressions more smoothly.When working with algebraic expressions:

• Substitute simplified radical expressions to keep equations neat.
• Use the properties of exponents and radicals carefully to combine like terms.
• Ensure to maintain the integrity of the expressions by verifying the simplified expressions through back-substitution if needed.

Simplifying radical expressions like square roots within algebraic contexts streamlines problem solving and enhances understanding of underlying concepts.