Problem 10 - Algebra 1: Concepts and Skills

Question

Use the power of a product property to simplify the expression. \((3 \cdot 4)^{3}\)

Step-by-Step Solution

Answer

The simplified form of the expression \((3 \cdot 4)^3\) is 1728.

Step 1: Identify the numbers and the exponent

The given expression is \((3 \cdot 4)^3\). So, \(a\) is 3, \(b\) is 4, and the exponent \(n\) is 3.

Step 2: Apply the power of a product property

The power of a product property states that \((ab)^n = a^n \cdot b^n\). So by applying that we get \(3^3 \cdot 4^3\).

Step 3: Simplify the expression

Now, simplify the expression by calculating the power of 3 and 4. We know that \(3^3 =27\) and \(4^3 =64\). So, \(3^3 \cdot 4^3 = 27 \cdot 64\).

Step 4: Calculate the product

Finally, calculate the product of 27 and 64, which is 1728.

Key Concepts

Exponents

Understanding exponents is critical in algebra. An exponent, also known as a power, represents the number of times a number (the base) is multiplied by itself. For example, the expression \(5^3\) means we should multiply 5 by itself 3 times: \((5 \times 5 \times 5 = 125\)).
Exponents can be more than just whole numbers; they can also be fractions, negative numbers, or even variables. However, in our given exercise, we're working with a positive whole number exponent, which simplifies our calculations and understanding of the concept.
The operation of raising a number to a power is so common in algebra that understanding how to manipulate these types of expressions is necessary for solving more complex problems.

Simplifying Algebraic Expressions

The goal of simplifying algebraic expressions is to make them as straightforward as possible. This often involves combining like terms, using the distributive property, and applying the rules of exponents.
In this case, simplifying involves applying the power of a product rule to an algebraic expression. Remember, the simpler the expression, the easier it is to understand or use in further calculations. Simplifying is not just about making an expression shorter; it’s about making it clearer and more accessible for further operations.
The power of a product rule is a handy tool in this simplification process. By using this property correctly, you break down complex exponent problems into more manageable pieces.

Properties of Exponents

The properties of exponents are the rules that govern how to handle expressions with exponents. One of these crucial properties is the power of a product property, which we've used in our example. It tells us that to raise a product to an exponent, we can raise each factor of the product to the exponent individually.
The power of a product property is expressed as \( (a \times b)^n = a^n \times b^n \). This simplifies calculations significantly. Other properties include the product of powers property (\( a^m \times a^n = a^{m+n} \) and the power of a power property (\( (a^m)^n = a^{m \times n} \)). Knowing how and when to use these properties can streamline solving algebraic problems.

Basic Algebra

At the foundation of all algebraic studies is basic algebra. It encompasses the fundamental principles and operations that allow us to manipulate equations and expressions. Concepts include understanding variables, constants, coefficients, and the conventions used to write algebraic expressions efficiently.
In our example, we have combined these foundational concepts: we encountered variables (the algebraic expressions), constants (the numbers 3 and 4), and an exponent. We then applied a specific algebraic property to simplify the expression. Basic algebra serves as the groundwork for all of this, and without a strong grasp of these fundamentals, progressing in mathematics would be challenging.