Problem 59 - College Algebra with Modeling and Visualization

Question

Exercises \(55-64:\) Use the power rules to simplify the expression. Use positive exponents to write your answer. $$ \left(4 y^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(64y^6\).

Step 1: Identify the Power Rule

The power rule in algebra states that when raising an expression that already has an exponent to another power, you multiply the exponents. This is expressed as \((a^m)^n = a^{m \cdot n}\). In this problem, the expression \((4y^2)^3\) should be simplified using this rule.

Step 2: Apply the Power Rule to Each Factor

Apply the power rule individually to each factor within the parentheses. First, apply the rule to the constant 4: \((4)^3\). Next, apply the rule to the variable expression \((y^2)^3\), which becomes \(y^{2 \cdot 3}\).

Step 3: Calculate the Powers

Calculate each power separately: \((4)^3 = 4 \times 4 \times 4 = 64\) and \(y^{2 \cdot 3} = y^6\). Therefore, these simplify to 64 and \(y^6\) respectively.

Step 4: Combine the Results

Combine the results from Step 3 to write the simplified expression: \(64y^6\). This is the final expression in its simplest form, and both the constant and exponent are included with positive exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves rewriting algebraic expressions in a more efficient or compact form without changing their value. During simplification, our goal is to reduce the complexity of the expression. This helps make further calculations and understanding much easier.
Typically, in algebraic expressions, simplification might involve:

• Combining like terms
• Applying mathematical operations, such as addition, subtraction, or multiplication
• Using rules of exponents, as seen in our exercise

By simplifying expressions, we potentially reveal more about the relationship between variables, and it's especially helpful in solving equations or inequalities. The original expression \((4y^2)^3\) was simplified to \(64y^6\) using exponentiation and multiplication. This indicates the straightforwardness and clarity that simplification can bring to algebraic problems.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations that combine to represent a specific value or set of values. They are the backbone of algebra, as they express relations and quantities that can be manipulated according to algebraic rules.
For example, in our given expression \((4y^2)^3\), we have:

• A coefficient: 4, which is a constant factor multiplying the variable part
• A variable: \(y\), representing an unknown quantity
• An exponent on the variable: 2, indicating how many times the variable is used as a factor

Expressions like these allow mathematicians and students alike to model real-world situations using symbolic and abstract representations. They help in finding solutions to problems by using substitution, graphing, or other algebraic techniques. Understanding the structure of algebraic expressions is crucial, as it enables more efficient problem solving by applying appropriate rules or methods.

Power Rule in Algebra

The power rule in algebra is a key exponentiation concept that simplifies expressions where an expression with an exponent is raised to another power. The "power of a power" rule states that \((a^m)^n = a^{m \cdot n}\), which means you multiply the exponents.
In our initial exercise, the expression \((4y^2)^3\) was simplified using this rule. Here’s a breakdown of how it applies:

• The base remains the same within the parentheses
• The exponents are multiplied together: the exponent of \(y^2\) is multiplied by 3, resulting in \(y^{2 \cdot 3}\) or \(y^6\)
• Additionally, constants like 4 also follow the rule: \((4)^3 = 64\)

Using the power rule effectively means recognizing when to apply this multiplication of exponents, which simplifies the process and can be applied universally across algebraic expressions. It's an indispensable tool for students when tackling problems involving exponential expressions.