Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{10 x^{6}}{5 x^{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^3 \).

Step 1: Identify Like Terms

In the expression \( \frac{10x^6}{5x^3} \), observe that both the numerator and the denominator have terms with powers of \( x \). The coefficients are 10 and 5.

Step 2: Simplify the Coefficient

Simplify the fraction by dividing the coefficients: \( \frac{10}{5} = 2 \).

Step 3: Apply the Property of Exponents

Simplify the power of \( x \): \( \frac{x^6}{x^3} = x^{6-3} = x^3 \). This follows the rule \( \frac{x^m}{x^n} = x^{m-n} \).

Step 4: Combine Terms

Combine the simplified coefficient and the simplified variable terms: \( 2x^3 \).

Key Concepts

Simplifying Expressions

Rational expressions can look complex at first glance, but with the right approach, simplifying them can be manageable. Simplifying expressions generally involves reducing a complex expression to its simplest form without changing its value. This makes them easier to understand and work with. To do this:

• Look for common factors or terms in the numerator and the denominator that can be simplified.
• Apply arithmetic operations to coefficients of like terms.
• Utilize mathematical properties to simplify variables, such as the properties of exponents.

In the example provided, \[ \frac{10x^6}{5x^3} \]at first, it appears elaborate, but by following these steps, it becomes very straightforward. The goal is to ensure the expression is as simple as possible, capturing all components but with minimal complexity.

Properties of Exponents

The properties of exponents are incredibly useful when working with algebraic expressions. These rules allow us to manipulate powers of numbers and variables effectively. One of the most common properties is the division of powers with the same base:

• When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator: \( \frac{x^m}{x^n} = x^{m-n} \).

In the expression \[ \frac{10x^6}{5x^3} \],the powers of \( x \) can be simplified by subtracting the exponent in the denominator from the exponent in the numerator: \( x^{6-3} = x^3 \).
This simplification is rooted in the property that when you divide like bases, their exponents subtract each other to give a new, simplified power.

Like Terms

Understanding like terms is a foundational concept in algebra, essential for properly simplifying expressions. Like terms are those that have the exact same variable raised to the same power. Their coefficients can be different, but the key point is their variable component:

• When simplifying, only like terms can be combined.

In the expression \[ \frac{10x^6}{5x^3} \],the terms \( 10x^6 \) and \( 5x^3 \) are considered 'like terms' in the sense that they both involve powers of the same base, \( x \), even though their exponent values differ.
By focusing on like terms, the expression can be reduced accurately and easily, as seen where the coefficients \( 10 \) and \( 5 \) are simplified, and the powers of \( x \) are adjusted according to exponent rules.