Problem 61 - Algebra A Combined Function

Question

\(\frac{15 x}{x+8} \cdot \frac{2 x+16}{3 x}\)

Step-by-Step Solution

Answer

The expression simplifies to 10.

Step 1: Simplify the Fractions

We start by simplifying each fraction independently. The first fraction is \(\frac{15x}{x+8}\). It can be simplified if we identify any common factor between the numerator \(15x\) and the denominator \(x+8\). However, there are no common factors here, so this fraction stays the same.The second fraction \(\frac{2x+16}{3x}\) can be simplified. We can factor out a 2 from the numerator:\[2x + 16 = 2(x + 8)\]Thus, the fraction becomes \(\frac{2(x+8)}{3x}\).

Step 2: Cancel Common Factors

Now, we multiply the fractions together:\[\frac{15x}{x+8} \cdot \frac{2(x+8)}{3x}\]We notice that \((x+8)\) appears in both one numerator and one denominator, so they cancel each other out:\[= \frac{15x \cdot 2}{3x}\]

Step 3: Simplify the Resulting Fraction

The fractions are now simplified to:\[\frac{30x}{3x}\]We can see that \(x\) is present in both the numerator and the denominator, so they cancel each other out. We also divide 30 and 3 by 3:\[\frac{30}{3} = 10\]So, the expression simplifies to 10.

Key Concepts

Algebra

Algebra is a branch of mathematics that focuses on symbols and the rules for manipulating these symbols. It is essentially about finding the unknowns and understanding how numbers relate to each other through operations. When dealing with algebraic expressions, it's common to encounter variables like 'x' that represent numbers. These variables can be part of expressions or equations, just like in this exercise.In our exercise, you can see how algebra helps manage expressions with variables. When we simplify expressions, we aim to make them easier to understand or work with by reducing them to their simplest form. This often involves recognizing patterns and using operations like addition, subtraction, multiplication, or division.Key Algebraic Operations:

• Factoring: Breaking down an expression into simpler parts that, when multiplied, give the original expression. For example, factoring out a common factor like in the expression \(2x + 16 = 2(x + 8)\).
• Canceling: Removing identical terms from the numerator and the denominator, which was done with \(x+8\) in the fractions.
• Simplification: Reducing an expression to its simplest form, important for clarity and ease of calculation, as we did in the final steps to reach the answer, 10.

Mathematics Education

Mathematics education involves teaching and learning math concepts in an effective way. Understanding how to work with fractions, especially in algebra, is a fundamental skill that is part of a math curriculum. It helps build a student’s capacity to think critically and solve problems. When teaching fraction simplification, it’s important to emphasize the core steps:

• Identifying factors and recognizing opportunities to simplify expressions.
• Understanding how operations affect terms by practicing examples with different complexities.
• Regular practice that includes both guided and independent problem-solving scenarios to strengthen skills.

By laying a strong foundation in these basic concepts, students find it easier to tackle more advanced mathematics topics. This approach is especially helpful when dealing with rational expressions as students progress in their math education.

Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator are polynomials. Understanding these is crucial in algebra because they frequently appear in equations and need to be simplified.Properties of Rational Expressions:

• Like regular fractions, rational expressions can often be simplified by canceling common factors.
• It’s crucial to identify and factor out any common terms in the polynomials of both the numerator and the denominator.
• Using these simplified forms of expressions can make solving equations much more manageable and reduce the likelihood of errors.

In our exercise, the rational expressions \(\frac{15x}{x+8} \) and \(\frac{2x+16}{3x}\) demonstrate these characteristics. Simplifying these expressions by canceling common factors and using factoring helps streamline solving complex problems in mathematics.