Problem 88 - Elementary Algebra

Question

For the following problems, expand the terms so that no exponents appear. $$ 7^{3} x^{2} $$

Step-by-Step Solution

Answer

Question: Expand the expression $$7^3 x^2$$. Answer: The expanded expression is $$343x^2$$.

Step 1: Expand the base 7 exponent

Calculate the value of $$7^3$$ by multiplying 7 by itself 3 times: $$7 \times 7 \times 7 = 343$$.

Step 2: Expand the x term

Since there is an exponent of 2 for the x term, we need to expand it as $$x \times x$$.

Step 3: Combine the expanded terms

Now that both terms have been expanded, we can combine them into their final form. The expanded expression is $$343(x \times x)$$, which can also be written as $$343x^2$$. The final expanded expression is $$343x^2$$.

Key Concepts

Exponential Expressions

Understanding exponential expressions is crucial in building a solid foundation in algebra. An exponential expression is a mathematical notation that indicates the operation of multiplying a base by itself a certain number of times. For example, the exponential expression \(7^3\) represents multiplying the base \(7\) by itself three times (\(7 \times 7 \times 7\)), resulting in \(343\).

It's essential to grasp that the exponent, also called the power, represents the number of times the base is used as a factor. When faced with a problem involving exponents, the first step is always to calculate the value of the base raised to the exponent, as seen in the textbook solution where \(7^3\) is expanded.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about reducing complexity while maintaining the integrity of the original expression. The process often involves expanding exponents, combining like terms, or applying the distributive property. To simplify the expression \(7^3 x^2\), for instance, you first expand the exponential terms, giving \(343\) for \(7^3\) and \(x \times x\) for \(x^2\).

The next step often includes eliminating parentheses by distributing coefficients or combining like terms, although in the case of \(343x^2\), the expression is already in its simplest form. The goal of simplification is to transform the expression into a form that is easier to interpret or use, especially in solving equations or further algebraic manipulations.

Algebraic Operations

Algebraic operations are the procedures that are performed on algebraic expressions, including addition, subtraction, multiplication, and division, similar to arithmetic but with variables included. When expanding exponents, such as with \(7^3 x^2\), multiplication is the primary operation employed. After the terms are expanded individually, they are often combined or manipulated through additional algebraic operations.

In our example, once we have \(343\) for \(7^3\) and \(x \times x\) for \(x^2\), these results can be combined through multiplication to form \(343x^2\). It is crucial to understand the rules that govern these operations, as they allow for the consistent and accurate simplification and manipulation of algebraic expressions.

Basic Algebra

Basic algebra introduces the fundamental principles and the language of algebra, including variables, constants, exponents, and the concept of an equation. It forms the basis for all subsequent topics in algebra. In the context of our example, \(7^3 x^2\), we observe the basic algebra concept of expanding exponential expressions and working with variables.

Grasping the concept that letters can represent numbers and can be manipulated using arithmetic operations is a foundational aspect of algebra. In performing the expansion of \(7^3 x^2\), we apply these basic algebraic skills to reach a more simplified and understandable form, \(343x^2\). It's essential for students to be comfortable with these basic algebra concepts to confidently approach more complex algebraic problems.