Problem 29 - Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt{242} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{242} \) is \( 11\sqrt{2} \).

Step 1: Identify Prime Factors

First, let's find the prime factorization of 242. Start by dividing 242 by the smallest prime number, which is 2. Since 242 is divisible by 2, divide it by 2 to get 121. Next, check if 121 is a prime number or find its prime factorization.

Step 2: Factorize 121

Notice that 121 is not a prime number. Instead, it is a perfect square because it equals 11 squared. Specifically, 121 can be factorized into 11 × 11.

Step 3: Combine Prime Factors

We now know the prime factorization of 242 is 2 × 11 × 11. Let's express this using powers of prime numbers: 242 = 2 × 11^2.

Step 4: Simplify the Radical

Now that we have found the factorization, simplify the square root of 242 using the property that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). In this case, \( \sqrt{242} = \sqrt{2 \times 11^2} = \sqrt{2} \times \sqrt{11^2} \).

Step 5: Simplify Further and Finalize

Since the square root of a square is the base of the square, \( \sqrt{11^2} = 11 \). Therefore, \( \sqrt{242} = \sqrt{2} \times 11 = 11\sqrt{2} \).

Key Concepts

Prime Factorization

Prime factorization is like finding the building blocks of a number. Every number is made up of prime numbers, which are numbers greater than 1 that can only be divided by 1 and themselves. To simplify radical expressions, you start by finding these prime factors. This means breaking down the number inside the radical into its prime components. For instance, with 242, you begin by dividing by the smallest prime number, 2, which results in 121.

• 242 is split into 2 and 121.
• 121 is not prime but is the result of 11 multiplied by 11.

So, the prime factorization of 242 is 2 and two 11s, or 2 × 11 × 11. Using powers, this is expressed as 2 × 11². Finding this factorization is the essential first step to simplifying any radical expression.

Square Root Properties

Square root properties help in simplifying radical expressions by breaking them into more manageable parts. The formula \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) allows us to separate the expression under the radical sign into its prime factors. This approach is beneficial because it splits the task into smaller, easier parts.
For example, with \( \sqrt{242} \), we know:

• \( 2 \times 11^2 \) are under the square root.
• This separates into \( \sqrt{2} \times \sqrt{11^2} \).

The square root of a square (like \( \sqrt{11^2} \)) simplifies directly to the base, which is 11 in this case. Thus, \( \sqrt{242} \) becomes \( 11 \sqrt{2} \). The use of square root properties is foundational to handle square roots effectively.

Perfect Square

A perfect square results from multiplying a number by itself. Identifying perfect squares is essential when working with square roots because they simplify directly and neatly. In our example, 121 is a perfect square because 11 times 11 equals 121.
Here’s a quick way to visualize perfect squares:

• 121 is \( 11 \times 11 \), which is the foundation of its perfect square status.
• Taking the square root of 121 simply returns its base, which is 11.

Spotting and working with perfect squares is a major time-saver in simplifying square roots. It reduces the expression significantly because the square root of a perfect square removes the square, leaving the base number unaltered and clean.