Problem 65 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{24}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{24}\) is \(2\sqrt{6}\).

Step 1: Finding the Prime Factors

To simplify any radical, the initial step is to find the prime factors of the number under the radical sign. Here, find the prime factors of 24. By doing prime factorization, 24 can be written as \(2^3 * 3\).

Step 2: Grouping the Prime Factors

The next step is to pair the factors in groups of two, as a square root always 'unpairs' a pair. Hence, group \(2^3\) as \(2^2 * 2\). This is because as a simplifying rule for radicals, \(\sqrt{a^2}\) is \(a\). Thus, \(2^2\) can be taken out of the square root, while 2 and 3 remain inside.

Step 3: Writing the Simplified Radical Expression

The last step is to write the simplified radical expression. From the previous step, \(2^2\) can be simply written as 2, and it comes out of the square root. Therefore, the simplified form of \(\sqrt{24}\) is \(2\sqrt{6}\).

Key Concepts

Prime Factorization

To start understanding how to simplify radical expressions, it's crucial to get familiar with prime factorization. Prime factorization is the process of breaking down a composite number into the product of its prime factors. Prime numbers are numbers greater than 1 that cannot be divided evenly by any other numbers except 1 and themselves.

For example, let's take the number 24. When we perform prime factorization on 24, we're looking for prime numbers that multiply together to give us 24. Starting with the smallest prime number, 2, we see that 24 can be divided by 2 to give us 12. Continuing this process, 12 is also divisible by 2, giving us 6. Then, 6 can be divided by 2 to get 3, which is a prime number itself. So, the prime factorization of 24 is expressed as \(2^3 \times 3\).

Understanding prime factorization is essential because it is the first step in simplifying radicals by revealing pairs of factors that can be simplified.

Square Roots

Square roots are a fundamental concept when dealing with radical expressions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9, denoted as \( \sqrt{9} \), is 3 because \(3 \times 3\) equal 9.

When evaluating square roots like \( \sqrt{24} \), we could guess and check to find numbers that multiply to 24, but this is often not practical for larger numbers or non-perfect squares. Instead, we use the results of prime factorization to work towards radical simplification. Besides understanding basic square roots, it's also important to know that the square root function 'undoes' the action of squaring a number. Consequently, any squared factor (like \(2^2\)) can be 'taken out' of the square root in simplified form.

Radical Simplification

With the concepts of prime factorization and square roots in hand, we can now move on to radical simplification. The main goal while simplifying a radical expression is to get the simplest form by eliminating the square root where possible, or reducing it to the smallest possible number. This is achieved by utilizing the factor pairs found during prime factorization.

In the case of \( \sqrt{24} \), we've found that the prime factorization is \(2^3 \times 3\). By grouping these factors into pairs (\(2^2 \times 2 \times 3\)), we can take the square root of these pairs. Since the square root of \(2^2\) is simply 2, we can 'take out' this square from beneath the radical sign, leaving us with \(2 \sqrt{6}\), where the radical is now simplified.

This simplified form not only looks cleaner, but it's also easier to work with in further calculations. Mastering radical simplification helps students solve more complex algebraic problems effectively.