Problem 3 - 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. $$\frac{8 x^{9}}{3 x^{7}}$$

Step-by-Step Solution

Answer

\( \frac{8x^2}{3} \)

Step 1: Analyze the Expression

The given expression is \( \frac{8x^9}{3x^7} \). This is a rational expression with both a numerator and a denominator containing powers of \( x \).

Step 2: Apply the Division Property of Exponents

The expression \( \frac{x^9}{x^7} \) can be simplified using the division property of exponents: \( x^a / x^b = x^{a-b} \). Here \( a = 9 \) and \( b = 7 \), so \( x^9 / x^7 = x^{9-7} = x^2 \).

Step 3: Simplify the Coefficient Fraction

The numerical coefficients in the expression are \( \frac{8}{3} \). This fraction cannot be simplified further, so it remains as \( \frac{8}{3} \).

Step 4: Combine Simplified Parts

Multiply the simplified results from the previous steps: \( \frac{8}{3} \times x^2 \). This gives \( \frac{8x^2}{3} \) as the simplified rational expression.

Key Concepts

Division Property of Exponents

When dealing with expressions that involve exponents, an important property is the division property of exponents. This property states that for any non-zero number, if you divide two expressions with the same base raised to powers, you subtract the powers. Mathematically, this is written as \( x^a / x^b = x^{a-b} \).
Let's take a closer look at how this rule applies to a specific application: simplifying the expression \( \frac{x^9}{x^7} \).
The bases here are both \( x \). So, applying the division property, we have \( x^{9-7} = x^2 \). This rule greatly simplifies expressions by reducing the exponents, making complex calculations more manageable.
Remember, this property only applies when the bases are the same, and both expressions must have exponents for subtraction to occur.

Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form without changing its value. In the context of rational expressions, simplifying involves two main components: the numerical coefficients and the variable parts.
With the expression \( \frac{8x^9}{3x^7} \), you've already used the division property of exponents to simplify the exponents part to \( x^2 \).
Next, we focus on the coefficients: \( \frac{8}{3} \). In this case, the fraction \( \frac{8}{3} \) is already in its simplest form as there are no common factors between 8 and 3 other than 1.
So, sometimes, your coefficients might simplify neatly into whole numbers, but sometimes they do not, and you have to keep them as a fraction. This is normal and an important part of working with rational expressions.

Numerator and Denominator Analysis

Understanding the roles of numerators and denominators is crucial in simplifying rational expressions. The numerator is the top part of the fraction, while the denominator is the bottom part. Each plays a specific role in determining the overall value of the fraction.
In our example, the original rational expression \( \frac{8x^9}{3x^7} \) has a numerator of \( 8x^9 \) and a denominator of \( 3x^7 \).
By looking at both parts separately, you can identify opportunities to simplify, such as using the division property of exponents for the \( x \) terms. After simplification, we end up with \( \frac{8x^2}{3} \), which combines the simplified numerator \( 8x^2 \) and the unchanged denominator 3.
This process of identifying and actively simplifying different parts of the expression is key to mastering rational expressions.