Problem 38 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ 3 y^{4} \cdot 6 y^{-4} \cdot y $$

Step-by-Step Solution

Answer

The simplified expression is \(18y\).

Step 1: Identify Like Terms

The expression has three components: \(3 y^4\), \(6 y^{-4}\), and \(y\). All components involve the variable \(y\) raised to some power. They can be combined using the product rule for exponents.

Step 2: Apply Product Rule for Exponents

The product rule for exponents states that \(a^m \cdot a^n = a^{m+n}\). So, we sum the exponents of all the \(y\) terms: \(4 + (-4) + 1\).

Step 3: Calculate the Exponent

Add the exponents: \(4 + (-4) + 1 = 1\). This means all the \(y\) terms combine to \(y^1\) or simply \(y\).

Step 4: Multiply Constants

Multiply the constant coefficients: \(3 \cdot 6 = 18\).

Step 5: Form the Simplified Expression

Combine the product of the constants (18) and the simplified \(y\) term: the simplified expression is \(18y\).

Key Concepts

Product Rule for Exponents

The product rule for exponents is a fundamental concept in algebra that helps us simplify expressions involving powers of the same base. When you multiply expressions that have the same base with different exponents, you use this rule to simplify it. It states:

• For any base \(a\), and exponents \(m\) and \(n\), \(a^m \cdot a^n = a^{m+n}\).

In other words, you **add the exponents** when multiplying like bases.
In the exercise given, we needed to simplify the expression \(3 y^4 \cdot 6 y^{-4} \cdot y\).
We focus on the terms with base \(y\), so we add the exponents \(4 + (-4) + 1\).
This addition results in \(y^1\), which we simply write as \(y\). This use of the product rule makes it much easier to handle complex algebraic expressions by effectively reducing the power terms.

Simplifying Expressions

Simplifying expressions is about rewriting them in the most straightforward form without changing their value.
Here, we took the expression \(3 y^4 \cdot 6 y^{-4} \cdot y\) and simplified it by performing a series of steps.

• First, identify that all the \(y\) terms could be combined using the product rule for exponents.
• Then, compute the new exponent by adding \(4 + (-4) + 1 = 1\).
• Finally, multiply the constant terms \(3\) and \(6\) to get \(18\).
• Combine these results to form the simplified expression: \(18y\).

Each step of simplifying removes complexity and leads to a clearer and more direct expression, making calculations easier and more manageable.
This process showcases how foundational algebraic techniques streamline the reduction of expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations that represent a specific value or relationship.
In our exercise, you encounter an expression \(3 y^4 \cdot 6 y^{-4} \cdot y\), which needs simplification.

• An algebraic expression like this consists of terms, which are individual parts separated by addition or subtraction (e.g., \(3 y^4\) is one term).
• Terms can contain constants (like \(3\) or \(6\)), variables (like \(y\)), and exponents (like \(4\) or \(-4\)).

The power of algebra resides in its ability to manipulate these expressions to solve equations, evaluate functions, and model real-world scenarios.
By understanding how exponents and variables work, especially through rules like the product rule, you can transform complex-looking expressions into simpler, more usable forms.
This knowledge is crucial for learning how to solve equations and navigate more advanced mathematical concepts.