Problem 75 - Algebra 2

Question

PREREQUISITE SKILL. Simplify. Assume that no variable equals 0. $$ \frac{x^{3}}{x} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^2 \).

Step 1: Identify the Rule for Exponents

We know that for any non-zero number \( a \) and integers \( m \) and \( n \), the quotient of powers rule states \( \frac{a^m}{a^n} = a^{m-n} \). This means we subtract the exponents when dividing like bases.

Step 2: Use the Rule on the Given Problem

Apply the quotient of powers rule to the expression \( \frac{x^3}{x} \). We have:\[ \frac{x^3}{x} = x^{3-1} \]

Step 3: Simplify the Exponent

Simplify the exponent in the expression \( x^{3-1} \), which becomes:\[ x^2 \]

Step 4: Present the Simplified Expression

The expression \( \frac{x^3}{x} \) simplifies to \( x^2 \). This is our final simplified result.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when simplifying algebraic expressions. Exponents, often called powers, denote how many times a number, known as the base, is multiplied by itself. If you see an expression like \( x^m \), it means \( x \) is multiplied by itself \( m \) times.

There are several key rules to remember:

• Product of Powers: When multiplying like bases, add their exponents. For example, \( x^a \times x^b = x^{a+b} \).
• Power of a Power: When raising an exponent to another power, multiply the exponents: \( (x^a)^b = x^{a \cdot b} \).
• Quotient of Powers: When dividing like bases, subtract their exponents: \( \frac{x^a}{x^b} = x^{a-b} \). This is the rule essential to the original exercise.
• Zero Exponent Rule: Any non-zero base raised to the zero power is 1: \( x^0 = 1 \).

By understanding these rules, you'll be able to systematically simplify various expressions involving exponents with ease.

Quotient of Powers

The quotient of powers rule is a fundamental part of exponent rules, which makes it very useful in algebra. When you divide two expressions that have the same base, you subtract the exponent in the denominator from the exponent in the numerator.

For example, consider the expression \( \frac{x^m}{x^n} \). By applying the quotient of powers rule, we get \( x^{m-n} \). It simplifies the expression by eliminating the common base exponentials.

In the given exercise, the expression \( \frac{x^3}{x} \) is simplified using this rule. Here, you take the power of 3 from the numerator and subtract the power of 1 from the denominator to get \( x^{3-1} = x^2 \).

This straightforward rule helps in breaking down complex algebraic fractions and makes it easier to handle algebra problems. It’s especially helpful in ensuring our expressions are simplified before solving or substituting values.

Prerequisite Algebra Skills

Before delving into exponent rules and the quotient of powers, it's helpful to establish a solid foundation in basic algebra skills. Knowing how to manage variables and constants is crucial.

Start by understanding the concepts of a variable and constant. A variable represents an unknown value, typically expressed by a letter such as \( x \) or \( y \), while a constant is a fixed numerical value.

Basic operations involving variables include addition, subtraction, multiplication, and division. Being familiar with these operations helps when tackling more complex expressions.

Another important skill related to simplifying expressions is factoring. Although not directly required in the given exercise, factoring helps to break down expressions, particularly when they don't appear to have common bases initially.

Acquiring these skills prepares you for more advanced topics in algebra, including better understanding and applying rules about exponents and simplifying expressions.