Problem 23 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{18} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Perfect Square

Identify the perfect square that divides 18 it is \(9 \times 2\). So, we rewrite the square root expression as \(\sqrt{9 \times 2}\).

Step 2: Apply the Square Root Property

Apply the property of square roots that says \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). This allows us to break down our expression into \(\sqrt{9} \times \sqrt{2}\).

Step 3: Simplify the Square Root

Finally, simplify the expression. The square root of 9 is 3, so our final simplified expression is \(3\sqrt{2}\).

Key Concepts

Understanding Perfect Squares

Knowing the concept of a perfect square is crucial when simplifying square roots. A perfect square is a number that can be expressed as the product of an integer with itself. For example, the number 16 is a perfect square because it equals 4 times 4. In the context of simplifying (\fracroot\(\sqrt{18}\)), recognizing that 9 is a perfect square within 18 (since 9 equals 3 times 3) is the first insightful step towards simplification.

When encountering a radical expression, identifying perfect squares allows us to break down the square roots into more manageable pieces. A handy list of common perfect squares includes 1, 4, 9, 16, 25, 36, and so on, which can be really helpful for quick reference. By having these memorized or easily accessible, you can effortlessly search for perfect square factors within the radicand (the number under the square root symbol).

Exploring Square Root Properties

Square root properties are essential tools in mathematics that help simplify radical expressions. One key property is that the square root of a product can be expressed as the product of the square roots of the individual factors. Mathematically, this is represented as \fracroot\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). It's important to note that this property is valid for non-negative numbers 'a' and 'b'.

In our exercise with (\fracroot\(\sqrt{18}\)), applying this property enables us to divide the radicand into a perfect square factor (9) and its partner (2), making simplification a systematic process. This property is incredibly useful and comes in handy across many areas of algebra and helps us in understanding the nature of square roots on a deeper level.

Radical Expression Simplification

Simplifying radical expressions involves reducing them to their simplest form using a series of algebraic steps. To simplify a square root, we look for perfect square factors, as we did with (\fracroot\(\sqrt{18}\)), split the radical using square root properties, then simplify each part.

An essential part of this process is to recognize that sometimes radicals cannot be completely freed of roots. In our example, the square root of 2 remains, represented as (\fracroot\(\sqrt{2}\)), because 2 is not a perfect square and cannot be simplified further. The goal of simplification is to pull out as much as we can in terms of whole numbers (like 3 from \fracroot\(\sqrt{9}\)), leaving the most straightforward radical expression possible. Understanding and practicing these steps ensure a robust ability to simplify any radical you encounter in your mathematical journey.