Problem 2 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{2} \cdot \sqrt{18} $$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Express each radical in terms of prime factors

Start by breaking down the numbers inside each square root into their prime factors. For \( \sqrt{2} \), the prime factor is \( 2 \) itself. For \( \sqrt{18} \), the prime factors are \( 18 = 2 \times 3^2 \). Thus, \( \sqrt{18} = \sqrt{2 \times 3^2} \).

Step 2: Apply the property of square roots

Use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to combine the square roots. Here, we have \( \sqrt{2} \cdot \sqrt{2 \times 3^2} = \sqrt{2 \cdot 2 \cdot 3^2} \).

Step 3: Simplify the expression inside the square root

Calculate the expression inside the square root: \( 2 \cdot 2 \cdot 3^2 = 4 \cdot 9 = 36 \). Thus, we have \( \sqrt{36} \).

Step 4: Simplify \( \sqrt{36} \)

The square root of 36 is 6, since \( 6^2 = 36 \). Therefore, \( \sqrt{36} = 6 \).

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This is a foundational tool in simplifying radicals because it allows us to break down numbers under square roots in terms of their simplest components.
For example, when we need to find the prime factorization of 18, we start by dividing it by the smallest prime, which is 2. Since 18 is even, we get:

• 18 ÷ 2 = 9
• 9 can further be divided into prime factors, using 3:
• 9 ÷ 3 = 3, so we have 9 = 3 × 3

Hence, the prime factors of 18 are 2 and 3 squared: 18 = 2 × 3².
When you write these down, the actual expression under the square root is simplified using its prime factors, as we did with \( \sqrt{18} = \sqrt{2 \times 3^2} \). This will make the simplification process of any radical expression much easier.

Property of Square Roots

One of the key properties of square roots that simplifies our calculations is \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
This property allows us to combine two square root numbers into one, which is essential in the simplification process.
In our example, we applied this property to \( \sqrt{2} \cdot \sqrt{18} \). With the prime factorization of 18, we got \( \sqrt{18} = \sqrt{2 \times 3^2} \).

• Using the property \( \sqrt{2} \cdot \sqrt{2 \times 3^2} = \sqrt{2 \cdot 2 \cdot 3^2} \)
• Here, we've combined the contents under a single square root: \( \sqrt{4 \cdot 9} \), allowing simplification to be easier.

This illustrates how leveraging properties of square roots can streamline the simplification process of complex expressions.

Expression Simplification

Expression simplification integrates everything we've learned - from prime factorization to applying the property of square roots. This process culminates in finding the simplest form of an expression.
For our example, after applying the property of square roots, we're left with the expression \( \sqrt{36} \).
Here, we recognize that 36 is a perfect square since it equals \( 6^2 \). Consequently, \( \sqrt{36} = 6 \).

• This result shows that the expression \( \sqrt{2} \cdot \sqrt{18} \) simplifies beautifully to 6.
• Each step - prime factorization, property application, and final calculation - aids in achieving this concise result.

Expression simplification not only assists in solving mathematical problems with clarity but also ensures solutions are as straightforward as possible.