Problem 10 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{44}$$

Step-by-Step Solution

Answer

\(\sqrt{44} = 2\sqrt{11}\)

Step 1: Prime Factorization

The first step is to find the prime factorization of the number under the radical. For \(44\), the prime factorization is \(2^2 \times 11\).

Step 2: Identify Perfect Squares

Next, identify any perfect squares in the factorization. In \(2^2 \times 11\), we can see that \(2^2\) is a perfect square.

Step 3: Simplify the Radical

Extract the square root of the perfect square from the radical. The square root of \(2^2\) is \(2\), so we can simplify \(\sqrt{44} = \sqrt{2^2 \times 11} = 2\sqrt{11}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1, which only have two divisors: 1 and themselves. When dealing with radicals, prime factorization helps us identify factors that can be simplified.
To use prime factorization, divide the number by the smallest possible prime number. Keep dividing until the only remaining factors are prime numbers.

• For example, start with 44: the smallest prime number to divide by is 2.
• 44 divided by 2 equals 22, which can also be divided by 2, resulting in 11.
• Since 11 is a prime number, we've completed the prime factorization: 44 = 2 × 2 × 11 or in exponential form, 2² × 11.

Now that we have the prime factorization of 44, we can look for perfect squares.

Perfect Squares

A perfect square is the product of an integer multiplied by itself. Recognizing perfect squares in the prime factorization allows us to simplify radicals effectively.
In our example of the number 44, we obtained the prime factorization 2² × 11.

• Here, 2² is a perfect square because it equals 4, which is 2 multiplied by 2.
• Perfect squares like 4, 9, 16, and 25 play a crucial role because they simplify to whole numbers when extracted from under the radical sign.

By identifying these perfect squares during prime factorization, we set up the foundation for simplifying the entire radical expression.

Square Root Extraction

The final step in simplifying radicals involves extracting the square root of any identified perfect squares. Extracting these components simplifies the expression under the radical. Given that perfect square identification is crucial, this step transforms the expression into its simplest form.
In our original exercise, we identified 2² as a perfect square from the prime factorization of 44.

• The square root of 2² is 2.
• So, the initial expression \(\sqrt{44}\) becomes \(2\sqrt{11}\).

This simplification makes it more manageable and easier to work with in further mathematical calculations or expressions. Remember, while 11 cannot be simplified further, identifying and extracting square roots enable us to present the simplest form of the radical.