Problem 33 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{-30 a^{2} b^{4}-35 a^{2} b^{3}-25 a^{2}}{-5 b^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: \(\frac{-30a^{2}b^{4} - 35a^{2}b^{3} - 25a^{2}}{-5b^{3}}\). Answer: \(6a^{2}b + 7a^{2} - 5a^{2}\frac{1}{b^{3}}\)

Step 1: Identify the terms in the numerator and the denominator

In the given expression, we have three terms in the numerator and one term in the denominator. Let's first write down these terms. Numerator terms: \(-30a^{2}b^{4}, -35a^{2}b^{3}, -25a^{2}\). Denominator term: \(-5b^{3}\)

Step 2: Divide each numerator term by the denominator term

Next, we'll divide each term in the numerator by the term in the denominator. \(\frac{-30a^{2}b^{4}}{-5b^{3}}\), \(\frac{-35a^{2}b^{3}}{-5b^{3}}\), \(\frac{-25a^{2}}{-5b^{3}}\)

Step 3: Simplify each fraction

Now, we'll simplify each fraction by cancelling out common factors. \(\frac{-30a^{2}b^{4}}{-5b^{3}} = 6a^{2}b\), \(\frac{-35a^{2}b^{3}}{-5b^{3}} = 7a^{2}\), \(\frac{-25a^{2}}{-5b^{3}} = 5a^{2}\frac{1}{b^{3}}\)

Step 4: Combine the simplified fractions

Finally, we'll combine the simplified fractions into one expression. \(6a^{2}b + 7a^{2} - 5a^{2}\frac{1}{b^{3}}\) And this is the simplified form of the given expression.

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions is a crucial skill in algebra. When simplifying, our goal is to make the expression as simple as possible, yet equivalent to the original form. To simplify an expression like the one given in the exercise, follow these steps:

• Identify and separate all terms within the expression — both in the numerator and the denominator.
• Divide each term in the numerator by the term in the denominator.
• Cancel out any common factors in each fraction.
• Lastly, combine these simplified terms to form the final expression.

By performing these steps, you make the expression easier to work with, whether you're solving equations, graphing, or substituting variables with numbers.

Numerator and Denominator

Understanding the numerator and denominator is fundamental when working with fractions. The numerator is the top part of a fraction, indicating the number of parts we have. The denominator is the bottom part, showing the total number of equal parts.

In algebraic fractions, terms in both the numerator and the denominator might contain variables. Take \(\frac{-30a^{2}b^{4}-35a^{2}b^{3}-25a^{2}}{-5b^{3}}\)as an example:

• The numerator comprises multiple terms, each expressing a part of the entire expression.
• The denominator is usually a single term in simpler exercises but can also be complex, depending on the problem's context.

When dividing terms by the denominator, simplify each fraction individually. This helps streamline the mathematical calculations, making them more manageable for further operations or interpretations.

Factoring in Algebra

Factoring is a method used to simplify algebraic expressions by expressing them as a product of their factors. This technique is particularly useful in identifying common components between two terms.

In the given exercise, each term in the numerator was divided by the denominator. Here, we factored out a number from both the numerator and the denominator, allowing us to simplify effectively. Here is how you can apply factoring:

• Identify any common factors shared by the numerator and the denominator. In our example, this is -5 from the denominator.
• Factor out these common elements, simplifying the terms.
• After canceling out these factors, it becomes simpler to operate on or further reduce the expression.

Through factoring, you can manage and solve algebraic problems more proficiently, leaving you with expressions that are often easier to interpret or solve.