Problem 67 - College Algebra Essentials

Question

Simplify the radical expressions in Exercises \(67-74\) if possible. $$\sqrt[3]{32}$$

Step-by-Step Solution

Answer

Therefore, the simplified form of \(\sqrt[3]{32}\) is \(2\sqrt[3]{4}\)

Step 1 - Identify Highest Perfect Cube

First, identify the highest perfect cube that evenly divides 32. That would be 8, since \(8*4=32\)

Step 2 - Rewrite The Expression

Next, rewrite the given cube root expression, breaking it into smaller parts. Rewrite \(\sqrt[3]{32}\) as \(\sqrt[3]{8*4}\)

Step 3 - Apply the Cube Root

Then, apply the cube root separately for 8 and for 4. The cube root of 8 is 2 because \(2^3 = 8\). However, 4 is not a perfect cube, meaning its cube root cannot be simplified any further.

Step 4 - Write the Final Expression

Finally, write your result as a simplified expression. So, \(\sqrt[3]{32} = 2 * \sqrt[3]{4}\)

Key Concepts

Perfect Cube

When simplifying radical expressions, recognizing a perfect cube is crucial. A perfect cube is a number that can be expressed as the product of an integer multiplied by itself twice more. In mathematical terms, a number \( n \) is a perfect cube if there exists an integer \( m \) such that \( n = m^3 \).
For example:

• \( 8 = 2^3 \)
• \( 27 = 3^3 \)
• \( 64 = 4^3 \)

Perfect cubes form the backbone for simplifying cube roots. They allow us to decompose a number into parts that can be easily simplified. In the exercise, we identify 8 as the highest perfect cube that divides 32. Recognizing perfect cubes helps break down expressions into more manageable pieces, simplifying the process significantly.

Cube Root

The concept of a cube root is essential when simplifying expressions involving third roots. The cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). It is typically denoted as \( \sqrt[3]{x} \).
In our exercise, the expression \( \sqrt[3]{32} \) involves finding two components: the cube root of a perfect cube (8) and a remaining non-perfect cube part (4). By decomposing 32 into \( 8 \times 4 \), we can directly take the cube root of 8, because 8 is a perfect cube. Simplifying this, \( \sqrt[3]{8} = 2 \). Meanwhile, 4 does not simplify nicely under a cube root, so it remains as \( \sqrt[3]{4} \).
Thus, knowing how to handle perfect cubes and cube roots empowers you to break down and simplify complex expressions.

Mathematical Expression Simplification

Simplifying mathematical expressions, especially those involving radicals, involves breaking down expressions into simpler parts. The prime focus is to express the radical in its simplest form by:

• Identifying perfect cubes (or squares for square roots).
• Rewriting expressions to separate perfect cube components and others.
• Simplifying cubes where possible.

This method was used in the step-by-step solution provided. Initially, we found 8 as a perfect cube of 32, allowing us to simplify \( \sqrt[3]{32} \) into \( 2 \times \sqrt[3]{4} \). Each step reduces the complexity, making the expression more manageable.
The key takeaway in simplifying such expressions is to identify components that can be reduced, keeping results tidy and computations more straightforward. Mastery of this skill enhances your mathematical fluency and problem-solving efficiency.