Problem 20 - Algebra 2

Question

Simplify. \(\sqrt[3]{54}\)

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{54}\) is \(3\sqrt[3]{2}\).

Step 1: Identify Cube Roots

The expression given is a cube root, \[ \sqrt[3]{54} \]Cube root seeks to identify what number, when multiplied by itself three times, gives 54.

Step 2: Factor the Expression Inside the Root

First, express 54 as a product of its prime factors: \[ 54 = 2 imes 3^3 \]This breaks down 54 into its prime elements, which assist in simplifying the cube root.

Step 3: Simplify the Cube Root Using Properties

Using the property of cube roots, we can simplify \[ \sqrt[3]{2 imes 3^3} \]by separating:\[ \sqrt[3]{2} imes \sqrt[3]{3^3} \]Since the cube root of \(3^3\) is simply 3, it becomes:\[ 3 \times \sqrt[3]{2} \]

Step 4: Simplification Conclusion

Since \(\sqrt[3]{2}\) is already in its simplest radical form, we conclude that the simplification process results in:\[ 3 \times \sqrt[3]{2} \]

Key Concepts

Prime Factorization

Prime factorization is a way of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, the prime factors of 54 are 2 and 3, which means 54 can be written as \( 54 = 2 \times 3^3 \).

• Find the smallest prime: Begin by dividing by the smallest prime number, which is 2. If not divisible, try the next smallest prime, like 3, 5, and so on.
• Repeat the process: Continue the process with the quotient until you're left with 1.
• List the factors: Once the division process ends, write all the prime numbers you have used as factors of the original number.

Recognizing the primes in our numerical expression helps in many scenarios, including simplifying radicals and identifying cube roots.

Simplifying Radicals

The process of simplifying radicals involves breaking down the number inside the radical sign (root) into a more manageable form. With cube roots, like \( \sqrt[3]{54} \), you use the number's prime factorization to break it down. This simplifies the computation of the cube root.

• Separate the prime factors: Using \( \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} \), you see the cube root can be split into individual radical expressions.
• Extract perfect cubes: Any factor that is a complete cube, such as \( 3^3 \), can be pulled outside of the radical, leaving the non-cube factors inside.
• Combine simplified parts: In this case, \( \sqrt[3]{3^3} \) resolves to \( 3 \), leaving \( 3 \times \sqrt[3]{2} \) as the solution.

This method not only simplifies but also provides a clean form that is easier to understand and work with.

Properties of Exponents

Understanding the properties of exponents is key to successfully simplifying expressions involving radicals.

• Power of a product: The property states that \((a \times b)^n = a^n \times b^n\), allowing the distribution of the exponent over the product.
• Nesting roots and exponents: When dealing with expressions like \( \sqrt[3]{3^3} \), realize the cube root and cube exponent "cancel" each other out, simplifying dramatically to \( 3 \).
• Simplification through division: Exponents can often be simplified through division; for example, the product \( 3^3 \) under a cube root simplifies directly because 3 and the root exponent are equal.

Mastering these properties helps in maneuvering through problems involving both radicals and exponents with ease.