Problem 65 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt[3]{16}\)

Step-by-Step Solution

Answer

The simplest radical form of \(\sqrt[3]{16}\) is \(2\sqrt[3]{2}\).

Step 1: Identify the index and radicand

The given radical expression is \(\sqrt[3]{16}\). Here, the index is 3, which indicates it's a cube root, and the radicand is 16.

Step 2: Factor the radicand

Find the prime factorization of the number inside the radical, which is 16. We can express 16 as a product of its prime factors: 16 = 2 × 2 × 2 × 2 = 2^4.

Step 3: Group the factors

Since we are working with a cube root (index 3), we need to group the factors into groups of three. From \(2^4\), we can take out \(2^3\) and leave one \(2\): \(2^3 \times 2\).

Step 4: Simplify the radical expression

Take the cube root of each group. The cube root of \(2^3\) is 2, because \(2^3\) simplifies out of the cube root. The expression becomes \(2\sqrt[3]{2}\).

Step 5: Write the simplest radical form

Combine the simplified terms to express the given cube root in its simplest radical form: \(\sqrt[3]{16} = 2\sqrt[3]{2}\).

Key Concepts

Cube Roots

Cube roots are a special type of root that deal with "cubing" a number, which means raising it to the power of three. When you see a cube root, like \( \sqrt[3]{x} \), it signifies the number that, when multiplied by itself twice, gives the original number \( x \). Understanding cube roots helps in simplifying expressions like \( \sqrt[3]{16} \).
Cube roots are used in various scenarios, especially when dealing with volume and geometry, as they help in determining dimensions of a shape.

• The cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
• The cube root symbol looks like a standard square root but with a tiny 3 above it, indicating it's a cube root.
• Instead of calculating directly, we often simplify cube roots using prime factorization.

Prime Factorization

Prime factorization is the process 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 16 are all 2s, because 16 can be expressed as \( 2^4 \).
This process is crucial when working with cube roots because it allows us to conveniently determine and simplify the radical expression.

• Finding the prime factorization helps to identify groups of factors, especially when simplifying cube roots.
• It involves breaking down the number step by step, dividing by the smallest prime number until what's left is a prime number.
• An easy way to approach it is by seeing how many times the number is divisible by 2, then 3, and so on.

Working with prime factorization is like peeling layers off an onion, making it easy to spot those factor groups that perfectly make up cubes.

Radical Expressions

Radical expressions involve roots of numbers and can sometimes seem tricky to simplify. A radical expression like \( \sqrt[3]{16} \) includes a radical symbol and what lies beneath it, known as the radicand.
Simplifying these expressions to their simplest form often requires a combination of techniques like recognizing cube roots and using prime factorization for clearer terms.

• The goal is to make the radical as simple as possible, often using smallest possible numbers.
• Radical expressions come in many forms and learning to manipulate them is a key skill in math.
• Understanding how to simplify these can also extend to higher levels of math, including algebra and calculus.

Mathematics Education

In mathematics education, learning about cube roots and radical expressions build a strong foundation for understanding more complex concepts. Simplifying radicals develops problem-solving skills and algebraic thinking, which are critical in a student's education.
Besides gaining mastery in mathematical calculations, understanding these concepts is about learning to apply them in practical and real-world scenarios.

• Applying these techniques, students can simplify heavy-looking math problems for easier solutions.
• Cube roots and radicals are connected to many branches of math, nurturing logic and analytical skills.
• Comprehending these concepts encourages curiosity and boosts students' confidence in dealing with numbers.

In short, mathematics education involving such topics not only helps in academic growth but also equips students with lifelong problem-solving skills.