Problem 62 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt[3]{64 y} $$

Step-by-Step Solution

Answer

The simplified expression is \(4 \sqrt[3]{y}\).

Step 1: Identify Parts of the Cube Root

The expression given is \( \sqrt[3]{64y} \). Here, we need to break down the cube root expression into its components. \(64\) is a constant, specifically \(4^3\), and \(y\) is a variable component.

Step 2: Simplify the Cube Root

We take the cube root of each component separately: \( \sqrt[3]{64} \) and \( \sqrt[3]{y} \). Since \(64 = 4^3\), the cube root of 64 is \(4\).

Step 3: Combine Simplified Parts

Now that we have \( \sqrt[3]{64} = 4 \), the simplified expression becomes \(4 \times \sqrt[3]{y}\).

Step 4: Complete the Simplification

Simplifying the entire expression yields \(4 \sqrt[3]{y}\) as our final simplified form.

Key Concepts

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, division, and roots). They are used in mathematics to represent a wide range of values and relationships. Understanding algebraic expressions is fundamental for solving various types of equations.

When simplifying expressions, such as in the given problem with a cube root, we break down the expression into more manageable parts. For example, in the expression \(\sqrt[3]{64y}\), we have a numerical constant (64) and a variable (y). Both components can be simplified separately, making complex expressions easier to handle.

• Algebraic expressions consist of constants, variables, and operations.
• Simplification helps to reduce complexity in solving equations.
• Decomposition of expressions aids in handling each part methodically.

variables in expressions

Variables are symbols, typically letters, that stand for unknown or changeable values. They give algebra flexibility and power, allowing mathematical models to describe different scenarios. In the expression \(\sqrt[3]{64y}\), the 'y' is a variable.

Variables can behave differently depending on the operations involved. In cube roots, for example, the variable stays under the root until further specific values or conditions are known, or more operations are performed.

• Variables allow generalization in algebraic expressions.
• They can represent unknown values, enabling problem-solving.
• Variables remain flexible in terms of operations they undergo.

mathematical constants

Mathematical constants are fixed values that do not change. They contrast with variables, which can vary. In an expression like \(\sqrt[3]{64y}\), 64 is a constant.

Constants often simplify to their roots when applicable. For instance, recognizing that 64 equals \(4^3\) allows us to simplify \(\sqrt[3]{64}\) to 4. Understanding this idea can be crucial in simplifying algebraic expressions.

• Constants have fixed values in mathematical expressions.
• Simplification arises from recognizing constants' properties.
• They bring predictability to expressions that include variables.