Problem 68 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-3 \sqrt[3]{54}\)

Step-by-Step Solution

Answer

The simplest radical form is \(-9 \sqrt[3]{2}\).

Step 1: Prime Factorization

First, find the prime factorization of 54.The number 54 can be factored into prime numbers as follows:1. Divide 54 by 2: \[ 54 \div 2 = 27 \] So, 54 = 2 × 27.2. Now, factor 27: 27 can be divided by 3: \[ 27 \div 3 = 9 \] So, 27 = 3 \times 9.3. Then, factor 9: 9 \div 3 = 3, so 9 = 3 \times 3.Thus, 54 can be expressed as:\[ 54 = 2 \times 3^3 \]

Step 2: Simplify the Radical

Since we are dealing with a cube root, look for groups of three identical factors.\[ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} \]Because there is a group of three 3's, we can take one 3 out of the cube root:\[ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3 \sqrt[3]{2} \]

Step 3: Apply the Coefficient

Now, apply the coefficient outside the radical.Recall the original expression: \(-3 \sqrt[3]{54}\)Replace \(\sqrt[3]{54}\) with the simplified radical form obtained:\(-3 \times 3 \sqrt[3]{2} = -9 \sqrt[3]{2}\)

Key Concepts

Understating Radical Expressions

When dealing with mathematics, especially algebra, you will encounter radical expressions. These are expressions that involve roots, such as square roots or cube roots. A radical sign (\(\sqrt{}\)) is used to denote the extraction of the root. In these cases, understanding what the radical represents is crucial.
The most basic form is the square root, denoted as \(\sqrt{\cdot}\), where you seek two identical factors of the number inside the radical. However, radicals can have different indices, like cube roots, which we'll discuss further later in this text. Remember:

• A radical expression simplifies when the radicand (the number inside the radical) is expressed using its factors.
• Simplifying reduces the expression to its most basic form while maintaining equality.

Grasping this concept helps in solving or simplifying problems involving radical expressions.

Understanding Cube Roots

Cube roots are a special type of radical expression, where the index or "degree" of the radical is three. It can be expressed as \(\sqrt[3]{x}\), meaning you're looking for a number that, when multiplied by itself twice more, gives the original number. For example, \(\sqrt[3]{27} = 3\) since \(3 \times 3 \times 3 = 27\).
Unlike square roots, cube roots of negative numbers can be negative since a negative number raised to an odd power remains negative. Let's consider \(-27\); \(\sqrt[3]{-27} = -3\) because \(-3 \times -3 \times -3 = -27\).

• Cube roots are particularly useful in simplifying expressions where the radicand is factored using its prime components.
• This understanding aids in revealing patterns or perfect cubes related to the expression's original form.

The Role of Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. For example, 54 can be broken down as \(2 \times 3^3\) because 54:2 gives 27, and further to \(3 \times 9\), and finally \(3 \times 3\). To simplify radicals effectively, knowing how to express the radicand as its prime factors is essential.
This helps identify "perfect powers" matching the radical's degree or index. For cube roots, this means finding groups of three identical numbers.

• Always begin by dividing by the smallest prime, usually 2, then proceed with larger primes.
• Continue until only prime numbers remain in the factorization.

This method is invaluable, not just for simplifying radicals, but also for various algebra and number theory applications.

Simplification Process of Radicals

Once the radicand is expressed through prime factorization, you can simplify the radical by using the properties of the specific root. In the case of cube roots, you look for triples in the factorization.
For example, with \(\sqrt[3]{54}\), the factorization was \(2 \times 3^3\). Notice the cube of 3? You can take 3 outside the radical:\[ \sqrt[3]{2 \times 3^3} = 3 \sqrt[3]{2} \]

• Breaking down into prime factors enables isolation of perfect powers that simplify the expression.
• Apply any coefficients, multiplying them by the simplified expression outside the radical.

Once you have simplified your radical expression, it becomes much easier to work with in subsequent mathematical operations. Each simplification step helps clarify and spotlight significant factors of the expression.