Problem 28 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{25 b^{4}}{55 b} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5b^3}{11} \).

Step 1: Identify the Common Factors

First, identify any common factors in the numerator and the denominator that can be canceled out. In the given rational expression \( \frac{25 b^4}{55 b} \), both the numerator and denominator are divisible by 5 and share the variable \( b \).

Step 2: Factor the Expression

Rewrite the expression to show these factors. This can be done by expressing both terms with their prime factors:\[\frac{25 b^4}{55 b} = \frac{5 imes 5 imes b^4}{5 imes 11 imes b}.\]

Step 3: Cancel the Common Factors

Cancel any common factors from both the numerator and the denominator. After simplification, remove one \( b \) from \( b^4 \) in the numerator to get \( b^3 \):\[\frac{5 imes 5 imes ot{b}^1 imes b^3}{5 imes 11 imes ot{b}^1} = \frac{5 imes b^3}{11}.\]

Step 4: Simplified Form

The expression is now simplified to \( \frac{5b^3}{11} \). Check to make sure no further simplification can be done. Since 5 and 11 have no common factors and the power of \( b \) is in its simplest form, the simplification is complete.

Key Concepts

Common Factors

When working with rational expressions, identifying common factors is a key step. A common factor is a number or variable that divides each term in an expression without any remainder. By finding these, you can reduce the expression to its simplest form.

• For numerical common factors, find the greatest number that can divide both the numerator and the denominator. For example, in the expression \( \frac{25 b^4}{55 b} \), the common numerical factor is 5.
• For algebraic expressions, look for common variables. In the given expression, both terms include the variable \( b \).

Recognizing and eliminating common factors can make complex problems simpler. This step is crucial for creating streamlined and workable expressions.

Prime Factorization

Prime factorization involves breaking down numbers into their basic building blocks—prime numbers. This makes it easier to see and work with common factors. Prime factorization expresses a number as a product of its prime factors. In our example: \[ 25 = 5 \times 5 \quad \text{and} \quad 55 = 5 \times 11 \]Prime factorization helps identify shared elements, facilitating simplification. Once prime factors are identified, removing common ones from the numerator and denominator becomes straightforward. Using prime factorization is like reducing a puzzle to simpler components. It gives a clear picture of what can be simplified and helps in organizing the terms for cancellation.

Cancellation

Cancellation is the process of removing identical factors from the numerator and the denominator to simplify an expression. After finding common factors and expressing the terms in their prime factorized form, you simplify by removing these common elements. In the process of simplifying \( \frac{25 b^4}{55 b} \), after factoring, you get:\[\frac{5 \times 5 \times b^4}{5 \times 11 \times b}\]

• Remove the common factor of 5 from both the numerator and the denominator.
• Cancel one \( b \) in the denominator with one \( b \) in the numerator, modifying \( b^4 \) to \( b^3 \).

This leaves you with \( \frac{5b^3}{11} \). Cancellation reduces expressions efficiently, focusing solely on distinct components.

Algebraic Expression Simplification

Simplifying algebraic expressions involves reducing them to the simplest form by applying different algebraic techniques. This includes identifying and cancelling common factors and utilizing prime factorization as discussed. The goal is to make the expression more manageable and easier to understand. Here are steps simplified:

• Identify and factorize all components.
• Remove any common factors through cancellation.
• Ensure the remaining expression cannot be simplified further.

For our expression \( \frac{25b^4}{55b} \), the simplified form is \( \frac{5b^3}{11} \). Each component is reduced to its least form, with no common factors left to cancel. By organizing expressions step-by-step, you can achieve clear and streamlined results. Simplifying algebraic expressions is like cleaning up a cluttered room, leaving only what's necessary.