Problem 91 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{3 x^{5}}{x^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3x \).

Step 1: Identify the Fraction Components

In the given expression \( \frac{3x^5}{x^4} \), identify the components of the fraction: the numerator is \( 3x^5 \) and the denominator is \( x^4 \).

Step 2: Apply the Law of Exponents

Recall the law of exponents which states that \( \frac{x^a}{x^b} = x^{a-b} \). Apply this to the expression by subtracting the exponent of \( x \) in the denominator from the exponent of \( x \) in the numerator: \( x^{5-4} = x^{1} \).

Step 3: Simplify the Expression

After applying the law of exponents, the expression becomes \( 3 \times x^1 \), which simplifies further to \( 3x \).

Key Concepts

Law of Exponents

When it comes to simplifying expressions involving exponents, the **Law of Exponents** is an invaluable tool. The key rule used in this context is that when dividing like bases, you subtract the exponents. This can be formulated as \( \frac{x^a}{x^b} = x^{a-b} \). So what happens here? Suppose you have \( \frac{3x^5}{x^4} \), you focus on the bases and their powers inside the fraction.
Here:- The base is \( x \).- The numerator has an exponent of 5.- The denominator has an exponent of 4.You subtract the exponent in the denominator (4) from the exponent in the numerator (5), resulting in \( x^{5-4} = x^1 \). This results in a significantly simpler expression. When simplified, \( x^1 \) is the same as \( x \). Understanding how to effectively use the Law of Exponents is crucial in algebraic simplifications.

Numerator and Denominator

In fractions, understanding the **Numerator** and **Denominator** is fundamental. The **Numerator** refers to the top part of a fraction, which indicates how many parts of a whole we are considering. Contrarily, the **Denominator** is the bottom part, showing how many parts make up a whole.
For example, in \( \frac{3x^5}{x^4} \):

• The numerator is \( 3x^5 \) which tells us we have 3 groups of \( x^5 \).
• The denominator is \( x^4 \) which divides the expression into groups of \( x^4 \).

Bringing attention to these two components is important because operations like simplification rely heavily on the interplay between the numerator and the denominator. In this exercise, when simplifying the entire fraction, you need to focus on the **like terms**, which are the powers of **x** found in both the numerator and denominator.

Fraction Simplification

**Fraction Simplification** is the process of reducing fractions to their simplest form. This means making the expression as concise as possible without changing its value. In algebra, simplifying fractions often involves reducing the powers of variables based on the principles like the Law of Exponents.
Take \( \frac{3x^5}{x^4} \) as your expression:

• Start by identifying the largest common component in the numerator and denominator. Here it's the base \( x \).
• You subtract the lesser exponent from the greater exponent. Using \( x^{5-4} = x^1 \).
• Multiply any remaining components, in this case, the 3, by the reduced variable to get the simplest form: \( 3x \).

Proper fraction simplification results in clearer expressions, making further math operations easier to handle. It is an essential skill, especially in algebra, as it avoids handling overly complex and cumbersome expressions.