Problem 62 - A Graphical Approach to College Algebra

Question

Factor, using the given common factor. Assume that all variables represent positive real numbers. $$y^{-5}-3 y^{-3} ; \quad y^{-5}$$

Step-by-Step Solution

Answer

The factored expression is \( y^{-5}(1 - 3y^2) \).

Step 1: Identify the Common Factor

First, observe that the expression to factor is \( y^{-5} - 3y^{-3} \), and the given common factor is \( y^{-5} \). This means \( y^{-5} \) is present in each term.

Step 2: Factor Out the Common Factor

Rewrite the expression by factoring \( y^{-5} \) out of each term. Begin with the first term:\[ y^{-5} = y^{-5} \times 1 \]For the second term, use the property \( a^m/a^n = a^{m-n} \):\[ 3y^{-3} = 3 \times y^{-5} \times y^{2} \]So the expression becomes:\[ y^{-5}(1 - 3y^2) \]

Step 3: Conclusion: Final Factored Form

Realize that this expression cannot be simplified further with respect to factoring. The final factored form of the expression is:\[ y^{-5}(1 - 3y^2) \]

Key Concepts

Common Factor

When dealing with algebraic expressions, identifying and factoring out a common factor is a fundamental step. A common factor is a term that is present in each element of the expression. Imagine it as a "hidden term" that both terms in an expression share.

In our example, the expression is \( y^{-5} - 3y^{-3} \) and the common factor given is \( y^{-5} \). This means both terms in the expression are divisible by \( y^{-5} \). Extracting the common factor simplifies the expression and makes further calculations easier.

• This process involves examining each part of the expression to see how it can be written as a multiple of this common factor.
  
• Once the common factor is identified, it can be factored out, effectively "pulling" it from each term to simplify the entire expression.

Recognizing and extracting common factors is not just about simplifying an equation but also setting the stage for solving complex problems more efficiently.

Exponent Rules

Understanding exponent rules is crucial when manipulating expressions with powers of variables. One key rule is the quotient of powers property, which states that \( \frac{a^m}{a^n} = a^{m-n} \). This rule is especially useful when factoring expressions like the one in our exercise.

In the provided problem, the second term \( 3y^{-3} \) can be rewritten in terms of the common factor \( y^{-5} \) by applying the quotient of powers rule:

• We start by expressing \( y^{-3} \) as \( y^{-5} \times y^2 \).
  
• This step shows how \( y^{-3} \) can be divided by \( y^{-5} \) to result in \( y^2 \), simplifying the term when combined with the common factor.

Mastering these exponent rules allows you to skillfully tackle a wide array of algebraic problems, particularly those involving powers and roots. Understanding how to apply these rules makes it much easier to simplify, expand, or factor expressions with exponents.

Algebraic Manipulations

Algebraic manipulations involve strategically rewriting expressions to simplify or solve them. It includes techniques such as factoring, expanding, and rearranging terms. In the problem at hand, we manipulate the expression \( y^{-5} - 3y^{-3} \) to its factored form. This process involves several key steps:

First, you identify the common factor and use it to transform the expression:

• Recognize that \( y^{-5} \) is a common factor and extract it from both terms of the expression.
  
• Rewrite the expression as \( y^{-5}(1 - 3y^2) \), factoring out \( y^{-5} \).

Such manipulations are essential in algebra for simplifying expressions, solving equations, and even graphing functions. They allow us to handle complex problems in a more manageable form, turning seemingly intricate expressions into clearer, more understandable representations.