Problem 40 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ y^{3} \cdot y^{-5} \cdot y^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( y^2 \).

Step 1: Identify the Product Rule

The product rule for exponents states that when multiplying two expressions with the same base, we add the exponents. Symbolically, if we have terms of the form \( a^m \cdot a^n \), we simplify this as \( a^{m+n} \).

Step 2: Apply the Product Rule

First, let's group the similar base terms: \( y^3 \cdot y^{-5} \cdot y^4 \). According to the product rule, we can write this as \( y^{3 + (-5) + 4} \).

Step 3: Simplify the Expression

Now simplify the exponent by performing the addition: \( 3 + (-5) + 4 = 2 \). Thus, the expression simplifies to \( y^2 \).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to the power of an exponent. It suggests repeated multiplication of the base. For example, the expression \( y^3 \) indicates that the base \( y \) is multiplied by itself three times, i.e., \( y \times y \times y \).
In the given exercise, we deal with the bases all being \( y \) but raised to different exponents: \( 3, -5, \text{ and } 4 \). The concept of negative exponents is also crucial here. A negative exponent, like \( y^{-5} \), implies division or the reciprocal, meaning that \( y^{-5} \) equals \( \frac{1}{y^5} \).
For any base raised to the zero power, for instance \( y^0 \), the result is always one. This understanding is central to simplifying expressions with exponents, particularly when using the product rule for exponents.

Algebraic Simplification

Algebraic simplification involves reducing mathematical expressions into their simplest form. This is done by combining like terms and using mathematical rules such as the product rule for exponents. In the context of this exercise, we used the product rule which is critical for simplifying expressions with the same base.
This rule states:

• When multiplying terms with the same base, add their exponents, i.e., \( a^m \cdot a^n = a^{m+n} \).
• This helps in combining several terms with different exponents into a single term.

In the step-by-step solution, all the terms have the base \( y \). By applying the product rule, the terms are combined into \( y^{3 + (-5) + 4} \), which simplifies to \( y^2 \). Thus, algebraic simplification leverages mathematical rules to condense expressions for easier understanding and handling.

Mathematics Education

Mathematics education aims to build foundational understanding through methods that break down complex processes. Simplifying expressions with exponents like in this exercise is part of the algebra curriculum often taught in middle or high school and is crucial for student progress in mathematics.
Understanding and applying the product rule is one of several important skills students develop. Skills acquired include:

• Logical reasoning and critical thinking by following a sequence of steps.
• Translating verbal rules into mathematical operations such as addition of exponents.
• Improving number sense by working with positive and negative integer values.

Engaging with these exercises prepares students for more advanced topics in algebra and calculus. Making the leap from theory to application builds confidence and an appreciation for math's structure and beauty.