Problem 74 - Intermediate Algebra

Question

Perform each division. \(\frac{22 a^{2} b^{2}-18 a^{2} b-52 a}{2 a b}\)

Step-by-Step Solution

Answer

The simplified expression is \(11ab - 9a - \frac{26}{b}\).

Step 1: Simplify Each Term in the Numerator

Write the given expression by separating the terms in the numerator: \[\frac{22a^2b^2}{2ab} - \frac{18a^2b}{2ab} - \frac{52a}{2ab}\] Simplify each fraction separately.

Step 2: Simplify First Term of the Numerator

Divide the term \(22a^2b^2\) by \(2ab\), which reduces as follows:\[\frac{22a^2b^2}{2ab} = \frac{22}{2} \times \frac{a^2}{a} \times \frac{b^2}{b} = 11ab\]

Step 3: Simplify Second Term of the Numerator

Divide the term \(18a^2b\) by \(2ab\), which simplifies to:\[\frac{18a^2b}{2ab} = \frac{18}{2} \times \frac{a^2}{a} \times \frac{b}{b} = 9a\]

Step 4: Simplify Third Term of the Numerator

Divide \(52a\) by \(2ab\), noting that \(b\) does not cancel:\[\frac{52a}{2ab} = \frac{52}{2} \times \frac{a}{a} \times \frac{1}{b} = 26 \times \frac{1}{b} = \frac{26}{b}\]

Step 5: Combine Simplified Terms

Now combine all of the simplified terms to write the complete simplified expression:\[11ab - 9a - \frac{26}{b}\]

Key Concepts

Numerator simplification

When dealing with algebraic expressions involving division, especially with polynomials, simplifying the numerator is a crucial step. It sets the stage for easier division and eventually helps in arriving at a simplified expression.
To start simplifying the numerator, separate each term and consider them individually. In this case, you must look at how each term in the numerator can be broken down when divided by the denominator.

• First, break down the expression \(22a^2b^2 - 18a^2b - 52a\) into separate fractions with the same denominator \(2ab\).
• Work with each fraction independently: \[\frac{22a^2b^2}{2ab} - \frac{18a^2b}{2ab} - \frac{52a}{2ab}\]
• Now, simplify each fraction by dividing coefficients and subtracting exponents where applicable.

By tackling each term of the numerator one by one, simplifying becomes much more manageable and less prone to mistakes.

Fraction division

Fraction division is a foundational skill in algebra, pivotal when simplifying complex algebraic expressions. It is crucial to understand how to properly divide fractions involving variables to proceed with expressions efficiently.
Start the process by understanding the basic rules of fraction division, which include:

• Dividing the coefficients directly — for example, \(\frac{22}{2}\) results in \(11\).
• Handling the variables separately. Here, for each variable, subtract the exponent of the divisor from the exponent of the numerator.

Let's consider our expression:

• Divide \(22a^2b^2\) by \(2ab\). Simplify the coefficient and apply exponent rules: \[\frac{22a^2b^2}{2ab} = 11ab\]
• Repeat this with remaining terms: \[\frac{18a^2b}{2ab} = 9a\] and \[\frac{52a}{2ab} = \frac{26}{b}\]

Remember, fraction division is about handling one piece at a time—making complex divisions simpler.

Algebraic expressions

Algebraic expressions can often appear complex, especially when they involve multiple variables and operations like division. To simplify these expressions, a structured approach is required.

• Break down expressions into smaller, manageable parts. This involves separating terms in a polynomial and handling them step by step.
• Understand the role of each component: terms, coefficients, and variables. Recognize patterns that allow simplification, like similar terms or common factors.
• Combine like terms only when necessary and only after simplification has been performed on each.

In our example, dealing with \(\frac{22a^2b^2-18a^2b-52a}{2ab}\) required taking each fraction independently and simplifying it. By knowing how to simplify algebraic expressions properly, you can transform what seems complex into a much more understandable and workable form, \(11ab - 9a - \frac{26}{b}\). This clarity helps in solving or further manipulating the expression efficiently.