Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. (Lesson 9.3) $$ \sqrt{18} $$

Step-by-Step Solution

Answer

\(\sqrt{18}\) simplifies to \(3\sqrt{2}\).

Step 1: Find the Prime Factors

Find the prime factors of 18. The prime factorization of 18 is \(2 \times 3 \times 3\). So, we can write \(\sqrt{18}\) as \(\sqrt{2 \times 3 \times 3}\)

Step 2: Group and Simplify

Group the prime factors in pairs. We have a pair of 3's. Write \(\sqrt{18}\) as \(\sqrt{2 \times 3^{2}}\) and take one 3 out from the radical.

Step 3: Apply the square root

Now, apply the square root to the pair of 3's. It becomes \(3\sqrt{2}\). We can't simplify it further as 2 doesn't have any pairs.

Key Concepts

Prime Factorization

Understanding prime factorization is fundamental when simplifying radicals. This process involves breaking down a number into its most basic building blocks, which are called prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, and 7.

To find the prime factorization of a number, continually divide the number by the smallest prime starting with 2 until you cannot divide anymore. For example, when we factorize 18, we start by dividing by 2 (the smallest prime number):

• 18 ÷ 2 = 9
• Next, we factor 9 by dividing by 3: 9 ÷ 3 = 3, and 3 is prime.

Thus, the prime factorization of 18 is 2 × 3 × 3 or 2 × 3². This prime factorization helps us transform the radical expression into a simpler form.

Square Root

The square root is a powerful mathematical operation that can simplify expressions, especially when dealing with radical expressions containing prime factors. The square root of a number is a value that, when multiplied by itself, gives the original number.

When simplifying square roots of a product of numbers, you can utilize the property: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property is particularly useful when applying it to prime factors.

In our example, after finding that 18 is equal to 2 × 3², we aim to pull out whole numbers from under the radical sign by identifying square number factors. Here, \(\sqrt{18}\) is transformed to \(\sqrt{2 \times 3^2}\). Recognizing that \(\sqrt{3^2} = 3\), we can simplify the term to \(3\sqrt{2}\), as 2 doesn't form a complete square pair and remains under the radical.

Radical Expressions

Radical expressions involve the root of numbers, often appearing as the square root, cube root, etc. Simplifying radicals is essential in various areas, including solving equations and graphing.

The goal in simplifying these expressions is to find any perfect square factors within the radicand (the number inside the radical) so that they can be simplified out of the radical.

• In the case of \(\sqrt{18}\), after breaking it into prime factors and rearranging as \(\sqrt{2 \times 3^2}\), the \(3^2\) is a perfect square inside the radical.

Recognizing such pairs is a key step as it lets you simplify to form expressions like \(3\sqrt{2}\), where 3 is taken out from under the radical, making the expression simpler and more manageable. Understanding how to manipulate and reduce these expressions is a critical skill in algebra and higher mathematics.