Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(5 x^{4}\left(x^{2}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(5x^6\).

Step 1: Identify the Base and Exponents

In the expression \(5 x^{4}\left(x^{2}\right)\), the base is \(x\) and the exponents are 4 and 2. The coefficient is 5.

Step 2: Apply the Law of Exponents

The law of exponents states that to multiply powers with the same base; you simply add the exponents. Therefore, we rewrite the expression as \(5x^{4+2}\) => \(5x^6\) .

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making the expression as straightforward as possible. In algebra, you often deal with terms that can be combined or adjusted using certain rules. This not only makes expressions easier to understand but also sets the stage for solving equations. Consider the expression given: in its unsimplified form, it looks like this:

• The original problem: \[ 5x^4(x^2) \]
• Its simpler form:\[ 5x^6 \]

The goal here is to reduce the clutter and combine terms in a way that maintains the expression's integrity while making it easier to work with in equations and further operations. The process of applying certain properties and laws is what transforms the expression into its simplest form.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operation symbols that represent a mathematical operation. For instance, the expression \[ 5x^4(x^2) \] combines a number (5) with a variable (x) raised to various powers.
Variables in these expressions act as placeholders that can take on any value. They allow the expression to represent a wide range of possible values. Terms in an algebraic expression are separated by additions, subtractions, or different layers of operations such as multiplication. Understanding the structure of an algebraic expression is crucial for applying the correct simplification techniques and effectively solving equations.

Laws of Exponents

The laws of exponents are a set of rules that help simplify expressions involving powers. These rules are invaluable when working with terms that have the same base. One key law is the product of powers property, which states:

• When multiplying like bases, you add the exponents: \[ x^m \times x^n = x^{m+n} \]

In our example, the expression was simplified using this law by combining the exponents of the base \( x \):

• Original exponents:\[ x^4 \] and \[ x^2 \]
• Simplified exponent:\[ x^{4+2} = x^6 \]

Using these laws correctly ensures that expressions are combined and reduced efficiently and accurately, making them simpler to work with in both arithmetic and algebraic problem-solving.