Problem 62 - College Algebra

Question

Simplify each expression. $$\sqrt[3]{64 y}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 4\sqrt[3]{y} \).

Step 1: Identify the Cube Root

In the expression \( \sqrt[3]{64y} \), you are asked to find the cube root of the term inside the radical symbol, which is \(64y\). This means you need to break down the expression inside to factors that can be easily simplified.

Step 2: Simplify the Cube Root of 64

Recognize that 64 is a perfect cube: \(64 = 4^3\). Therefore, \( \sqrt[3]{64} = 4 \) because the cube root of 64 is 4.

Step 3: Simplify the Cube Root of y

Because \( y \) is a separate factor under the cube root, it remains inside the radical as \( \sqrt[3]{y} \), since it's not a perfect cube itself.

Step 4: Combine Results

Combine the results from Step 2 and Step 3. The expression \( \sqrt[3]{64y} \) simplifies to \( 4\sqrt[3]{y} \). The 4 is the simplified portion of the number, and \( \sqrt[3]{y} \) remains as it is.

Key Concepts

Simplifying Radical Expressions

When we talk about simplifying radical expressions, we're focusing on making the expression easier to manage by breaking it down into its simplest form. A radical expression involves roots, such as square roots or cube roots. The goal is to identify factors that can be easily worked with and removed from under the radical.

Here's a brief overview of the process:

• Identify the type of root involved, such as a cube root.
• Factor the expression inside the radical to spot any perfect powers.
• Isolate and simplify these factors using their roots.
• Combine the simplified results for the final expression.

Simplifying a cube root, like in the problem with \(\sqrt[3]{64y}\), allows us to reduce complex numbers or algebraic expressions into a more workable form. Thus, identifying perfect cubes and other factors becomes crucial in this process.

Perfect Cubes

Perfect cubes are numbers or expressions that can be expressed as some integer, multiplied by itself three times. Recognizing these numbers is key when simplifying cube roots, as they allow us to eliminate the radical.

Common examples include:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)

When simplifying an expression like \(\sqrt[3]{64y}\), recognizing that 64 is a perfect cube (\(4^3\)) is critical. This knowledge helps us immediately simplify \(\sqrt[3]{64}\) to 4, reducing the complexity of the expression.

Algebraic Expressions

Algebraic expressions represent a combination of numbers, variables, and operators (such as +, -, *, and /). Understanding how to manipulate these expressions is fundamental in solving many math problems.

Key components involve:

• Variables, like \(y\), that can take on various values.
• Coefficients, numbers that multiply the variables.
• Operators that dictate the operations to perform on the variables and coefficients.

In the sequence of simplifying \(\sqrt[3]{64y}\), understanding how to manage the variable \(y\) as part of the expression is essential. Since \(y\) is not a perfect cube, it remains inside the radical after the simplification, demonstrating the interplay of variables in algebraic expressions. Recognizing and working with these parts enable you to confidently handle and simplify expressions.