Problem 13 - Elementary and Intermediate Algebra

Question

Divide the monomials. See Example 1. $$ \frac{x^{5}}{x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^3 \).

Step 1: Identify the Monomials

We are given the expression \( \frac{x^5}{x^2} \) to simplify.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \), where \( m \) and \( n \) are integers and \( a \) is a non-zero number. Apply this rule to divide the monomials: \( \frac{x^5}{x^2} = x^{5-2} \).

Step 3: Simplify the Exponent

Simplify the expression from the previous step: \( x^{5-2} = x^3 \).

Key Concepts

Understanding Monomials

A monomial is a type of algebraic expression that represents a single term. It can include numbers, letters, or a combination of both raised to positive whole number exponents. For instance, in the problem \( \frac{x^5}{x^2} \), both \( x^5 \) and \( x^2 \) are monomials. Each monomial comprises a coefficient (which can be a number like 2 or 3), a variable (such as \( x \), \( y \), etc.), and an exponent which is a whole number.
When dealing with monomials, it's crucial to recognize that you are working with a single coherent term rather than a sum or a difference of terms. The simplicity of monomials makes them easier to manipulate, especially when applying algebraic rules like the quotient rule. Familiarity with monomials sets the stage for more complex expressions and operations.

Exploring the Quotient Rule

The quotient rule is a fundamental principle in algebra that deals with the division of exponential expressions. When you divide two monomials with the same base, you subtract the exponent of the divisor from the exponent of the dividend. The formula is given by:

• \( \frac{a^m}{a^n} = a^{m-n} \)

In our example, \( \frac{x^5}{x^2} \), the base is \( x \), \( m = 5 \), and \( n = 2 \). Applying the quotient rule results in:

• \( x^{5-2} = x^3 \)

This rule simplifies calculations by reducing the need for lengthy multiplication or division. Remember, this rule only applies when the bases of the expressions are identical and the exponents are positive integers. Practicing the quotient rule helps in developing a stronger grasp of exponent manipulation and leads to efficiently simplifying algebraic fractions.

Grasping the Concept of Exponents

An exponent indicates how many times a base number is multiplied by itself. For example, \( x^5 \) means \( x \) is multiplied by itself five times. Exponents are an efficient way to express repeated multiplication. They appear in many areas of algebra and higher mathematics.
Here are a few key points about exponents:

• \( x^1 \) is simply \( x \).
• \( x^0 \) equals 1, regardless of the base, as long as the base is not zero.
• Negative exponents represent reciprocals, i.e., \( x^{-3} = \frac{1}{x^3} \).

Understanding exponents is essential for manipulating algebraic expressions, as they often simplify complex operations. Learning these rules will give you a strong foundation to tackle more complicated algebraic calculations. By using the properties of exponents, tasks like simplifying \( \frac{x^5}{x^2} \) are straightforward and intuitive.