Problem 46 - Elementary Algebra

Question

\(\sqrt{18}\)

Step-by-Step Solution

Answer

\(\sqrt{18} = 3\sqrt{2}\)

Step 1: Identify the Problem

The goal is to simplify the square root of 18.

Step 2: Prime Factorization

First, find the prime factors of 18. Break it down: 18 can be written as 2 * 9. Then further break down 9 as 3 * 3. Thus, the prime factors of 18 are 2 and 3^2.

Step 3: Simplify Under the Radical

Using the prime factors, rewrite the square root: \(\sqrt{18} = \sqrt{2 \cdot 3^2}\)

Step 4: Apply the Square Root Property

Apply the property \(\sqrt{a\cdot b} = \sqrt{a} \cdot \sqrt{b}\), which gives \(\sqrt{2 \cdot 3^2} = \sqrt{2} \cdot \sqrt{3^2}\). Since \(\sqrt{3^2} = 3\), we have \(\sqrt{2} \cdot 3\)

Step 5: Simplified Form

The simplified form is \(\3\sqrt{2}\). Thus, \(\sqrt{18} = 3\sqrt{2}\).

Key Concepts

prime factorization

Prime factorization is the process of breaking down a composite number into a product of its prime numbers. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, the number 18 can be broken down as:

• First, divide by the smallest prime number, 2: 18 ÷ 2 = 9
• Next, 9 can be divided by 3, another prime: 9 ÷ 3 = 3
• Lastly, 3 itself is a prime number.

So, 18 can be written as 2 * 3 * 3 or 2 * 3². Knowing the prime factors helps us when we want to simplify square roots.

square root properties

Understanding the properties of square roots makes it easier to simplify them. One important property is: \(\sqrt{a * b} = \sqrt{a} * \sqrt{b} \) This property allows us to break down a square root into more manageable parts. Another key property is that the square root of a square of a number returns the original number: \(\sqrt{a²} = a\). For example, using these properties for \(\sqrt{18}\), and knowing that 18 = 2 * 9, helps us simplify: \(\sqrt{18} = \sqrt{2 * 3²} = \sqrt{2} * \sqrt{3²} = \sqrt{2} * 3.\)

simplifying radicals

Simplifying a square root, or radical, means breaking it down into its simplest form. Let's use the properties of square roots and prime factorization.

First, find the prime factorization of the number inside the radical: 18 = 2 * 3². Rewrite the square root using this factorization: \(\sqrt{18} = \sqrt{2 * 3²}\).

Next, apply the square root property: \(\sqrt{2 * 3²} \) becomes \(\sqrt{2} * \sqrt{3²}\). Since the square root of \(\sqrt{3²} \) is simply 3, we have: \(\sqrt{2} * 3\).

So, the simplified form is \(\3 \sqrt{2} \). This is how you simplify \(\sqrt{18}\) to \(\3 \sqrt{2}\).