Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{x^{2} \cdot x^{n}}{x^{3} \cdot x^{n}}\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \(x^{-1}\).

Step 1: Apply the Rules of Exponents

First apply the rule of multiplying exponents with the same base: \(x^{m} \cdot x^{n} = x^{m+n}\). Hence, \(\frac{x^{2} \cdot x^{n}}{x^{3} \cdot x^{n}}\) simplifies to \(\frac{x^{2+n}}{x^{3+n}}\).

Step 2: Simplify Using Exponent Subtraction Rule

Next, apply the rule of dividing exponents with the same base: \(x^{m} / x^{n} = x^{m-n}\). Hence, \(\frac{x^{2+n}}{x^{3+n}}\) simplifies to \(x^{(2+n)-(3+n)}\).

Step 3: Simplify the Exponent

Finally, simplify the expression inside the parentheses in the exponent: \(x^{(2+n)-(3+n)}\) simplifies to \(x^{-1}\).

Key Concepts

Simplification

Simplifying algebraic expressions refers to the process of making them more manageable and easier to work with. This involves reducing the expression to its simplest form by performing arithmetic operations and combining like terms.

• When simplifying an expression like \(\frac{x^{2} \cdot x^{n}}{x^{3} \cdot x^{n}}\), the goal is to express it in the simplest way possible while retaining its original value.
• In this exercise, simplifying the expression involved using exponent rules to combine terms and reduce the complexity.
• By understanding these rules, one can effectively simplify complex algebraic expressions, making calculations much easier.

Simplification not only helps in solving specific problems, but also in understanding the behavior of expressions under various operations.

Exponent Rules

Exponent rules are essential when performing algebraic manipulations involving terms with powers. These rules simplify the process of multiplying or dividing expressions with the same base.
Some common rules include:

• Product of Powers: When multiplying like bases, you add their exponents: \(x^{m} \cdot x^{n} = x^{m+n}\).
• Quotient of Powers: When dividing like bases, you subtract the exponents: \(x^{m} / x^{n} = x^{m-n}\).

In the given exercise, the expression \(\frac{x^{2} \cdot x^{n}}{x^{3} \cdot x^{n}}\) was simplified by applying these fundamental exponent rules.
Understanding these rules is crucial, as they provide a systematic means to manipulate and solve algebraic expressions that contain exponents.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. These expressions can range from simple to very complex.

• They often involve factors such as exponents, which indicate how many times a number or variable is multiplied by itself.
• In the given exercise, variables were raised to various powers, which formed the basis for the simplification process.
• Algebraic expressions like \(\frac{x^{2} \cdot x^{n}}{x^{3} \cdot x^{n}}\) require understanding of exponent rules to be accurately simplified.

Grasping how to interpret and manipulate algebraic expressions is fundamental not only in algebra but also in advanced fields of mathematics. It equips you with the skills necessary to tackle equations, inequalities, and functions with confidence.