Problem 49 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{24 y}\)

Step-by-Step Solution

Answer

The simplest radical form is \(2 \sqrt[3]{3y}\).

Step 1: Break Down the Expression

The expression to simplify is \( \sqrt[3]{24y} \). Start by factoring 24 into its prime components: 24 = 2 * 2 * 2 * 3.

Step 2: Simplify Inside the Radical

Since the expression is a cube root, look for a group of three identical factors inside the radical. Notice 24 = \(2 \times 2 \times 2 \times 3\), and this can be paired as \( (2^3) \times 3 \).

Step 3: Extract Perfect Cubes

Identify the perfect cube \(2^3\) in the factorization. Extract \(2\) from the cube root, since \( \sqrt[3]{2^3} = 2 \). The expression outside the cube root is 2, and inside is \( \sqrt[3]{3y} \).

Step 4: Write the Simplified Expression

Combine the extracted factor with the remaining expression under the cube root. The expression simplifies to: \( 2 \sqrt[3]{3y} \).

Key Concepts

Cube Roots

Understanding cube roots is essential when simplifying radical expressions involving cubes. A cube root is similar to a square root but involves finding a value that, when multiplied by itself three times, gives the original number.
For example, the cube root of 8 is denoted as \( \sqrt[3]{8} \), which equals 2 because \( 2 \, \times \, 2 \, \times \, 2 = 8 \). Cube roots are particularly important in these types of problems because they help us simplify expressions by reducing them to their simplest form.
Simplifying cube roots requires finding perfect cubes, which are numbers like 1, 8, 27, and 64. These numbers have the property that their cube roots are integers, making it easier to work with radical expressions.

Radical Expressions

Radical expressions include any mathematical expressions that contain a root, such as a square root, a cube root, or even higher roots. In the context of simplifying radicals, the goal is to break down the number under the root to identify any portions that can be simplified.
For example, consider \( \sqrt[3]{24y} \). First, we identify the parts of 24 that can be grouped into a perfect cube. By factoring 24, we get \( 24 = 2 \times 2 \times 2 \times 3 \). The idea is to find groupings, such as \( 2^3 \), to simplify the expression.
Simplifying involves understanding which factors can be taken outside the radical. In this case, the perfect cube \( 2^3 \) allows us to extract 2, leading to \( 2\sqrt[3]{3y} \). This step is crucial in working with radical expressions as it simplifies them into more manageable terms.

Prime Factorization

Prime factorization is breaking down a number into its smallest 'building blocks', which are prime numbers. Every integer greater than 1 is either a prime number itself or a product of prime numbers, which makes this concept fundamentally important in simplifying expressions.
When simplifying radicals or other expressions, prime factorization enables us to identify patterns or groupings that represent powers of numbers, such as squares and cubes.
For example, the number 24 can be factorized into \( 2 \times 2 \times 2 \times 3 \) which is \( 2^3 \times 3 \).

• This decomposition allows us to spot the perfect cube within the factors easily.
• Once identified, these can be extracted from the radical expression using properties of roots.

Recognizing these prime factors at an early stage usually simplifies the process, thereby making it easier to deal with complex radical expressions.